See Shapiro and Teukolsky, Chapter 15Magnetic Accretion onto Neutron StarsA crucial difference between neutron stars and black holes is that neutron stars can havean intrinsic magnetic field. As discussed before, that field can be extremely strong, so strongthat the flow of ionized gas is channeled by the field. This produces the phenomenon ofaccretion-powered pulsars, and is also critical in the evolution of millisecond pulsars. We’llstart by looking at the interaction of the field with the inflowing matter, then investigateits consequences.Alfv´en radiusAs a first calculation, let’s see if we can figure out the characteristic radius at whichmagnetic stresses dominate the flow in the accretion disk. The region inside this radius iscalled the magnetosphere. This involves comparing quantities. Ask class: what quantitiesshould we compare to see if the magnetic field can channel the flow? As a first guess, wecould try energy density. The magnetic energy density is B2/8π, and the kinetic energydensity of the matter is12ρv2, where ρ is the density and v is the typical velocity. Specifically,suppose that the magnetic field is dipolar, so that B = µ/r3, and that the matter moves inspherical radial free fall, so that v = vff=q2GM/r. By continuity, ρ =˙M/(4πvffr2).Before solving this equation, note the radial dependences. The magnetic energy densitygoes as r−6, whereas the material energy density goes as r−5/2. The magnetic stresses thusincrease much more steeply with decreasing radius than the material stresses do. Therefore,generically one expects that far from the star, material stresses must dominate. Close tothe star, magnetic stresses will dominate if the field is strong enough; for B = 1012G, themagnetic stresses at the stellar surface are orders of magnitude stronger than the materialstresses, so there is some radius where the two balance approximately. This radius issometimes called the Alfv´en radius, and isrA=õ42GM˙M2!1/7= 3.2 × 108˙M−2/717µ4/730ÃMM¯!−1/7cm . (1)A magnetic moment of µ = 1030G cm3gives a surface field of about 1012G, so this istypical of neutron stars in high-mass X-ray binaries. Since the radius of a neutron star isR ≈ 106cm, the accretion flow onto a strongly magnetized neutron star is dominated bythe magnetic field.Caveat: the preceeding derivation gives an approximate value for this stress balanceradius, not an exact one. For example, one would really be interested in nearly circularflow, so the numbers would change a bit. Also, to be careful one should really comparethe rφ components of the stress, because this is relevant for nearly circular flow that ismoving in slowly. However, the ultimate answer for the radius is close to what we derived,for the simple reason that the radial dependence of the magnetic stress is much strongerthan that of the material stress, so a little change in the radius and the ratio of stresseschanges dramatically. Another subtlety is that there is, of course, not a sharp transitionfrom matter-dominated to field-dominated. Still, this is good enough for a start.Ask class: suppose that the field is quadrupolar instead of dipolar. Would you expectthe transition to be narrower or broader than in the dipolar case? Narrower, because theradial dependence of the field stresses is steeper.The details of how the plasma in the disk actually hooks onto the magnetosphere arecomplicated. It may be that magnetic Rayleigh-Taylor instabilities play a role, or it couldbe magnetic reconnection. It’s sketchy, and a problem is that observation of the relevantsources (accretion-powered pulsars) can’t tell us about the specific plasma physics.Spinup by magnetic accretionJust as with any accretion from a disk, angular momentum is accreted. Let’s startby imagining that the star is not rotating. Then, to a reasonable approximation, theangular momentum accreted per time (i.e., the torque) is just the accretion rate timesthe specific angular momentum at rA, or N ≈˙M√GMrA. Therefore, the star spins up.Now, suppose that the star has spun up to a frequency equal to the orbital frequency atrA: ωs=qGM/r3A. Ask class: what effect does further accretion have on the star, giventhat the specific angular momentum at rAis unchanged (by assumption)? The star’s spinfrequency is not changed, but its angular momentum goes up (this is possible becausethe moment of inertia increases). To understand this, let’s define the corotation radiusrco. At rco, the Keplerian angular velocity equals the spin angular velocity of the star:ωspin=qGM/r3co. Material in Keplerian orbits outside rcothat interacts with the star viathe magnetic field exerts a braking torque on the star, whereas material in Keplerian orbitsinside rcothat interacts with the star speeds the star up. When rco≈ rA, the two roughlycancel each other (in reality the radius at which the torques balance is slightly differentfrom rA).Now suppose that the star is spinning much faster than the Keplerian frequency at rA.Ask class: qualitatively, what should happen? In a rough sense, one expects that the starwill be slowed down by the coupling with the matter, because of a “drag” exerted at theinteraction radius. It may also be (and this is a topic of current debate!) that the matteris flung out by this interaction, as if the field was like a propeller (hence this is called the“propeller effect” or, more generally, a centrifugal barrier). If so, one expects that the massaccretion rate would drop drastically if the propeller phase were entered. Ask class: inwhat circumstance might one imagine that such a phase would be entered in a real system?Many sources are transients, and from the above equations it is clear that if˙M changesrapidly, so will rA. There are some cases in which evidence for a propeller phase has beenclaimed, due to sharp dropoff in luminosity, but this is difficult in practice because theluminosity is low as such a phase is approached.That means that, if there is time and˙M and B are constant, one expects that magneticaccretion will tend to make the star spin at the Keplerian frequency at rA. Ask class:how can we find out how long it will take until the star spins at roughly this equilibriumfrequency? Figure out the angular momentum that needs to be accreted, then determinethe time necessary from the specific angular momentum at rAand the mass accretionrate. Suppose you have an equilibrium frequency of 1 rad s−1, which is typical for starsin