Magnetospheres of stars (and giant planets) Interaction of wind and magnetic field Jiří Krtička Masaryk University Introduction Interaction of wind and magnetic field Stellar winds of hot star are ionized wind flows along the magnetic field-lines. Idealized MHD (a -> oo): dB dB — = V x (v x B) => V x (v x B) = 0 for — = 0. dt v 1 v 1 dt About 10% of OB stars have strong (measurable) magnetic fields with surface strengths of the order of 0.1 - 10 kG (Auriere at al. 2007, Romanyuk 2007): magnetic O stars (e.g., 91 Ori C, HD 191612, Donati et al. 2002, 2006) and chemically peculiar stars (He-rich and He-poor: a Ori E, CU Vir, Oksala et al. 2015, Kochukhov et al. 2014). What is the influence of the magnetic field on the outflow? l Wind magnetic confinement parameter 77 Magnetic confinement parameter 77 The effect of the the magnetic field is given by the ratio of the magnetic field density and wind kinetic energy density: 1i = 3r-l- 2 With wind mass-loss rate M = 47vr2pv and replacing B = 6eq with equatorial field strength, r = R* with the stellar radius, and v = with the terminal velocity we derive V* — .-, 5 My, 00 which is wind magnetic confinement parameter (ud-Doula & Owocki 2002). Weak confinement 77* < 1 77* < 1: the wind energy density dominates over the magnetic field energy density. The magnetic fields opens up. The wind flows radially. (ud-Doula & Owocki 2002) Strong confinement 77* > 1 77* > 1: the magnetic field energy density dominates over the wind energy density. Wind trapped by the magnetic field, collision of wind flow from opposite footpoints of magnetic loops: magnetically confined wind shocks (Babel & Montmerle 1997). Arrows denote infall (the flow is not stable, ud-Doula & Owocki 2002). 4 Magnetic confinement parameter 77* in real stars v* = Mv, 00 • strong winds in massive stars (e.g., HD 191612): for M « 10~6 M0 yr-1 the magnetic field of the order of 100 G is needed for strong confinement • weak winds in cool stars (e.g., Sun): for M ^ 10-14 M0 yr-1 the magnetic field of the order of 1 G is enough for strong confinement (e.g., corona) 5 Alfven radius Reshuffling the confinement parameter in terms of Alfven speed v/\ 1 p2 JL V >2 8?r ^ _ _ _ _ \pv2 v2 v2 M2 inverse of the confinement parameter represents the square of the Alfvenic Mach number M/\ = v/v/\. The radial dependence (with dipolar field with B ~ r~3 and v —>* v^) i-B2 r2 B2r2 , ^pv2 r2 Mv =>- the wind speed overcomes the Alfven speed (rj = Ma = 1) at Alfven radius /?a, from fits of numerical simulation (ud-Doula et al. 2008) R. a R 0.3 + (77, + 0.25)1/4 * 6 Wind quenching by magnetic confinement Closed loops: M/\ < 1 everywhere. The last closed loop: M/\ = 1 at the loop apex. Wind leaves the star on open loops that have M/\ > 1 at the loop apex significant reduction of the net mass-loss rate: wind quenching. (Babel & Montmerle 1997, ud-Doula & Owocki 2008, Petit et al. 2017) 7 Wind quenching by magnetic confinement From the equation for the dipolar field r = /?apexSin#2, where Rapex is the apex radius and 9 is the colatitude, equating Rapex = Ra gives that the magnetic field is open for 9 < 9/\ given by 9 a = a resin R * Numerical simulations (ud-Doula et al. 2008) give slightly lower maximum radius of closed magnetic loop as Rc ^ /?* + 0.7(/?a — ^*) The escaping wind fraction M f^c I R fB = ---=/ sin 6»d(9 = 1 -cos(9r = 1 * MB=o Jo V Rc is of the order of 0.01 — 0.1 for strong fields. Magnetic hot stars may be progenitors of massive binary black holes (Petit et al. 2017), which were detected as gravitational wave sources (Abbott et al. 2016). 8 he effect on mass flux Magnetic field influences also the mass flux from the stellar photosphere. The spherically symmetric stationary equation of motion has the form of (v • V)v = -gj + g-radf, where we neglected the gas pressure term, g* = GM(1 — l~)/r2, l~ is the Eddington factor, and gra(j is the acceleration due to the lines, _ 1 keFQ f dv/dr\a ^rad 1-a c \pcQKeJ Here a and Q are line force parameters (Castor, Abbott, & Klein 1975, Gayley 1995) and F is the total flux at radius r. 9 In magnetic field the wind flows along the magnetic field along the direction s tilted by an angle 9b with respect to the vertical direction Equation of motion has in the plane parallel case the form of (v • V)(s • v) = -g*(s • z) + g-rad(s • z). With v = vss we derive (v • V)(s • v) = vs(s • V)(vs) = = MbKs^j + ~ V%vslte = I^bVs^ without horizontal velocity variations = 0). The radiative force is (s • z)grad = ——(s • z)^— -L-Z I — a c \ pcQKe J _ pB keFQ (PBdvs/dz I — a c \ pcQKe _ I^b keFQ (p2BVsdvs/dz 1 — a c \ mrcQne where the radial mass flux is mr = pbPvs. he effect on mass flux Substituting w' = (vs/g*)dvs/dz the cw-a l+w' according to the value of C (rhr). Maximum mass-loss rate for tanget of the radiative force equal to one: aCcjUga^a_1 = 1, from the momentum equation wfc = a/(l — a) and Q = a~a(l - a)a-1/^-2a and mr = /4^cak, where aticak is the mass-flux for jib = 1 (Owocki & ud-Doula 2004). li he effect of stellar rotation Kepler corotation radius Non-degenerate stars: in the strong confinement limit 77* ^> 1 the magnetosphere rotates as a rigid body close to the star. Kepler corrotation radius: the velocity of the corotationg matter equal to the Kepler (orbital) velocity: 1/3 GM _ _ Vrot_ (GM i^ = qrk = i^rk^rk = {^ In terms of orbital rotation fraction W: w = Vre* = _}U ^Rk= W-2/3R Vorb . I GM The material may be centrifugally supported for r > Rk 12 Dynamical versus centrifugal magnetospheres dynamical magnetosphere R/\ < Rk'. material trapped in the magnetosphere falls back to the surface (as in the case without rotation) centrifugal magnetosphere /?a > Rk- material is supported against gravity by the magnetically enforced corotation: material can stay in equilibrium in magnetosphere 13 Centrifugal magnetospheres The effective potential in the frame that corotates with the star 0 71/4 7C/2 37C/4 K Minima of the potential along the field line accumulation of matter (Townsend & Owocki 2005, Prvak 2011). 14 Centrifugal magnetospheres The effective potential in the frame that corotates with the star 4>efF=----Q2r2s\n29 r 2 in dimensionless units £ = r/R^ is V|/eff=^0eff=_I_Ie2sin20 GM £ 2 s Minima of the potential along the field line accumulation of matter (Townsend & Owocki 2005, Prvak 2011). hydrostatic hydrostatic equilibrium along the field line 4>efF -P = Po I ~P-^=— rigidly rotating magnetosphere (RMM) model 14 Showcase of RMM model: a Ori E: magnetosphere 0 = 0.0 0 = 0.2 0 = 0.4 (Oskala et al. 2015) 15 Showcase of RMM model: a Ori E: Ha line Observations ~\-r Oh ARRM Model (i = 75°) ARRM Model (i = 85° i-1-1-1- t f J J_I_I_L t-r J_I_I_L 0.25 ■ 0.20 - 0.15 - 0.10 0.05 ]0.00 -0.05 -0.10 -0.15 x fa 'cc CD -1000-500 0 500 1000 -1000-500 0 500 1000 -1000-500 0 500 1000 Velocity v (kms-1) Velocity v (kms-1) Velocity v (kms-1) (Oskala et al. 2015) 16 Showcase of RMM model: a Ori E: optical variability phase (p More complex field (tilted hexapole) Rotating magnetospheres: observables Many quantities are variable in magnetic stars with rotating magnetospheres: • Ha line emission (Sundqvist et al. 2012) • visual photometry (Townsend et al. 2005, Wade et al. 2011, Krticka 2016) • wind line profiles (Marcolino et al. 2013) • continuum polarization (Carciofi et al. 2013) • X-ray emission (Naze et al. 2016) • radio emission 19 Massive star in the plot of R^/Rk vs. luminosity < O .........1......... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 •v. • ".........1 . . ° a*4 ; □ £ • Ha em. • Ha abs. 1 □ ......i.........i..... 5 4 log (L./L0) 0 stars: typically dynamical magnetospheres with Ha emission due to the winds. B stars: dynamical vs. centrifugal magnetospheres with Ha emission due to the circumstellar environment (Petit et al. 2013). 20 Open questions • 3D MHD for non-axisymmetric cases (non-aligned magnetic fields) • MHD modelling of higher order multipoles • magnetospheric leakage (Owocki & Cranmer 2018) 21 Magnetic braking Magnetic field in equatorial plane (spherical coordinates) Frozen field condition for the component: 1 d V x (v x B) = 0 => -— [r(vrB^ - v^Br)\ = 0 By integrating and evaluating the constant at R where vr < v^: r(yrB^ - v$Br) = -R*QBri0. Assuming B = 6(r) the Maxwell's equation V • B = 0 gives TB = r2Br = /?2er,0 and B. 4> B, Close to the star: ^ rft, Br ^> B^\ solid body rotation. In outer regions: < rfi, negative B^, no co-rotation. 22 Wind equations in equatorial plane (spherical coordinates) Momentum equation (neglecting the gas pressure) 1 p(v • V)V = -pgj + pgradf + —(V X B) X B 47T has in the azimuthal direction the form of d Br d From pvvr2 = const, and r26r = const, we derive d£ dr where = 0. C = rv---. Angular momentum carried by the gas and by the magnetic field is constant. 23 Angular momentum loss Inserting B^ = B^v^ — rQ.)/vr into equation for C we derive vjC 2 v = rQ2-2~' where the (radial) Alfven speed is v/\ = Br/'y/Aitp. Azimuthal velocity finite at the Alfven radius R/\ where vr = v/\\ jC = RfaUj. Angular momentum (per unit of mass) behaves as if the star rotates as a solid body out to the Alfven radius (Weber & Davis 1967, Lamers & Cassinelli 2000). Angular momentum due to the wind and magnetic stresses. 24 The spin down time Stellar angular momentum loss J = MRffi gives with the stellar angular momentum J = rjMR^Q. the spin down time 7~spin — ■ J r]MR 2 * MRl For solar-type stars M = /W(ft) and /?a (1972) law holds: ft ~ y/i. 30 /?a(ft) and the Skumanich "D O q3 Q. "o5 c o CO o 0 (>Hyi 71 UMa * EK Dra Skumanich 0 1 age [10 yr] Ribas et al. (2005) 25 Spin down time in massive stars With /?a ~ R^rjl^ the spin down time in massive stars is from the numerical simulations (ud-Doula et al. 2009) 1 1/2 M spin ^ 1.1 x 108yr 1 Mr: 0 a lkG r 0 oo 108 cm s-1 M lO-9/W0yr -l U I I I I I I I I I I I I I I I I I I I I I I I I l/J 3 I Spin down time of a Ori E 1.34 Myr (Townsend et al. 2010) agrees with 1.7 Myr predicted from theoretical wind models (Krticka 2014). o.o 0.01 0.00 -0.01 s o ii--U .....$ I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 4 6 £/1000 8 10 12 26 Radio emission & planets Radio emission Alfveri, surface \ \ Auroral radio emission _ \ Auroral radio emission ' ' / / / / Collision of wind streams from oposite hemispheres: X-rays. Close to the Alfven radius: magnetic reconnection, generation of fast electrons. Electrons propagate toward the star: gyrosynchrotron radiation. Collision of electrons withe stellar surface: reflected electron cause beamed auroral radio emission due to the coherent electron cyclotron maser emission. (Leto et al. 2018) Magnetospheres of giant planets The rapid rotation of the gas giant planets, Jupiter and Saturn, leads to the formation of magnetodisc regions in their magnetospheric environments. In these regions, relatively cold plasma is confined towards the equatorial regions, and the magnetic field generated by the azimuthal (ring) current adds to the planetary dipole, forming radially distended field lines near the equatorial plane. The ensuing force balance in the equatorial magnetodisc is strongly influenced by centrifugal stress and by the thermal pressure of hot ion populations, whose thermal energy is large compared to the magnitude of their centrifugal potential energy. The sources of plasma for the Jovian and Kronian magnetospheres are the respective satellites lo (a volcanic moon) and Enceladus (an icy moon). (Achilleos et al. 2015) 28 Magnetospheres of giant planets Magnetodisks: rapid rotation + small angle between the magnetic and rotational axes. Directly observed by spacecrafts (e.g., Voyager, Galileo, Cassini). (Achilleos et al. 2015) Saturn: auroral emission from the boundary between open and closed field lines due to interaction of magnetosphere and solar wind. (Bunce et al. 2008) Jupiter: radio emission due to synchrotron radiation of relativistic electrons. (Chang & Davis 1962) 29 Magnetospheres of acreting stars (Hartmann 2016) 30