Why do stars have such weak magnetic fields?

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Why do stars have such weak magnetic fields?

ContentsIntroduction: the flux problem in star formationDisconnection of star from surroundingsMagnetic reorganisation in radiative starsAnd in convective protostars


“Magnetic flux problem” in star formationFor cloud to collapse, gravitational (negative) energy mustovercome the positive energiesIn adiabatic collapse, both gravitational and magnetic energy growas R - 1 .[Thermal and rotational kinetic grow faster, so need to radiate heatand transport angular momentum outwards.]On scales > 1000 AU, observations show Egrav ~ EmagIn stars, Egrav >> EmagWhat happens to all the flux?Proposition: as star becomes disconnected from parent cloud,field is destroyed by instability[Not to be confused with the other flux problem: how to get fromEgrav < Emag on large scales to Egrav > Emag and so allow collapse inthe first place]


Observations of cores5


Observations of cores


Hourglass forming in simulationsKudoh & Basu 2007


“Magnetic flux problem” in star formationOft-heard solution−magnetic field somehow avoids being dragged inwardsDiffusion in disc: matter accretes, slipping past field lines(e.g. van Ballegooijen 1989)Diffusion allows matter to collapse across field lines(e.g. Mestel & Spitzer 1956)Opinions differ as to how efficient these processes are, and how they work


ContentsIntroduction: the flux problem in star formationDisconnection of star from surroundingsMagnetic reorganisation in radiative starsAnd in convective protostars


Hourglass picture:flux frozen intocore as it collapsesLater on, field linesdisconnect (as wesee in pre-MSstars, after mainaccretion phase)Disconnectionhappens at latestwhen super-Alfvénic windappearsacbdthen field canreorganiseinternally10


ContentsIntroduction: the flux problem in star formationDisconnection of star from surroundingsMagnetic reorganisation in radiative starsAnd in convective protostars


Reconnection in radiativestars as means to reduce fluxImagine star contains a large poloidal fluxInstability --> reconnection--> magnetic energy destroyedPoloidal field:star can be thought of asfluid of aligned bar magnets(Markey & Tayler 1973,74;Wright 1973;Flowers & Ruderman 1977)Simulations confirm this........


End result of reconnectionReconnection/relaxation ends when equilibrium is reachedThis equilibrium is called a “fossil field” and lasts for ~Hubble timeHow to predict strength of fossil field?


Formation of equilibria:helicity conservation and strength of equilibrium fieldMagnetic helicity definition:H ≡∫A · B dV where B = ∇×AH is perfectly conserved in a fluid of infinite conductivity (Woltjer 1958)H has units length x energy, so is roughly conserved while energy is dissipatedon small scales


Formation of equilibria:helicity conservation and strength of equilibrium fieldMagnetic helicity definition:H ≡∫A · B dV where B = ∇×AH is perfectly conserved in a fluid of infinite conductivity (Woltjer 1958)H has units length x energy, so is roughly conserved while energy is dissipatedon small scalesA given equilibrium type n has an associated dimensionless length scale λnH eq≈ λ n R and H eq ≈ H init therefore E eq ≈ H initE eq λ n REnergy of final equilibrium (and strength of ‘fossil field’) can be predicted if weknow the initial helicity. Energy (or poloidal flux) of initial field is not relevant.Helicity can be thought of asH ∼ Φ pol Φ tor


Finding equilibria: numerical methodsConcept−−−Make numerical model of radiative star with arbitrary B-fieldStar has stable entropy profileFollow evolution of field as it relaxes into an equilibrium


Finding equilibria: numerical methodsConcept−−−Make numerical model of radiative star with arbitrary B-fieldStar has stable entropy profileFollow evolution of field as it relaxes into an equilibriumFirst results− Simple axisymmetric equilibria are found (Braithwaite & Spruit 2004)17


Most basic equilibriumBraithwaite & Nordlund 2006


Comparison with observationsField topology of α 2 CVn, an A-type main-sequence star(‏imaging (Kochukhov et al. 2002, using Zeeman-Doppler


Energy and helicity in simulationsBraithwaite 2010Helicity falls only by~20% during relaxation(depends on resolution)Energy of equilibriumdepends on helicityLog energy against log helicitySolid lines represent equilibrium fieldStraight dashed line shows:H eqE eq≈ λ n R


Summary so farPoloidal flux during formation is not relevant parameterfor strength of fields seen in upper-main-sequenceHelicity is relevant parameter:Net toroidal flux and therefore helicity might be verysmall...?In many hourglassmodels, H = 0Some asymmetryrequired for non-zerohelicityH ∼ Φ pol Φ tor


ContentsIntroduction: the flux problem in star formationDisconnection of star from surroundingsMagnetic reorganisation in radiative starsAnd in convective protostars


Isentropic stars: buoyant loss of magnetic fieldReconnection & reorganisation towards an equilibriumproceeds at Alfvén speed or somewhat lessDeuterium burning drives convection and star becomesisentropicIn isentropic star, magnetic field provides buoyancy because itgives pressure without massRegions of higher field strength are driven upwardsThis motion also happens at Alfvén speed:∆ρρ∼ ∆PP ∼ B224πP ∼ v2 Ac 2 sandH p ≈ c2 sgandF buoy ∼ l 3 g∆ρ ∼ ρl 2 v 2 buoy ∼ F dragConvective motion is not relevant for a super-equipartitionmagnetic field⇒v buoyv A≈( lH p) 1/2


Isentropic stars: buoyant loss of magnetic fieldEnd result: original magnetic field is lost intoatmosphereReplaced by dynamo-driven field, at most atequipartition with convective motionConsequence: magnetic field independent of initialconditions


Simulations of relaxation in isentropic star“Star in box” simulation; star is polytrope ofindex n=3/2 with ideal gas e.o.s.Actual convectionnot modelled


Simulations of relaxation in isentropic star“Star in box” simulation; star is polytrope ofindex n=3/2 with ideal gas e.o.s.Actual convectionnot modelledEnergy andhelicity is lostinto atmosphereBraithwaite 2011,Duez & Braithwaite, in prep.


SummaryPoloidal flux during formation is not relevant parameter forstrength of fields seen in upper-main-sequenceHelicity is relevant parameter:H ∼ Φ pol Φ torNet toroidal flux and therefore helicity might be very small...?Asymmetry required for non-zero helicityHelicity can be destroyed efficiently while star remainsisentropic, i.e. convectively unstableTherefore lower-main-sequence stars naturally lose all theiroriginal helicity; magnetic properties are independent of initialconditionsIn short: any unwanted excess flux can be destroyed inprotostar before stellar surface even becomes visibleIf star accretes from disc: slippage mechanism still required,but it has extra “incentive” because star will not accept flux

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