The role of magnetic fields for planetary formation
aa r X i v : . [ a s t r o - ph . E P ] M a y Cosmic Magnetic Fields: from Planets, to Stars and GalaxiesProceedings IAU Symposium No. 259, 2009K.G. Strassmeier, A.G. Kosovichev & J.E. Beckman, eds. c (cid:13) The role of magnetic fieldsfor planetary formation
Anders Johansen Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlandsemail: [email protected]
Abstract.
The role of magnetic fields for the formation of planets is reviewed. Protoplanetary disc tur-bulence driven by the magnetorotational instability has a huge influence on the early stagesof planet formation. Small dust grains are transported both vertically and radially in the discby turbulent diffusion, counteracting sedimentation to the mid-plane and transporting crys-talline material from the hot inner disc to the outer parts. The conclusion from recent effortsto measure the turbulent diffusion coefficient of magnetorotational turbulence is that turbulentdiffusion of small particles is much stronger than naively thought. Larger particles – pebbles,rocks and boulders – get trapped in long-lived high pressure regions that arise spontaneouslyat large scales in the turbulent flow. These gas high pressures, in geostrophic balance with asub-Keplerian/super-Keplerian zonal flow envelope, are excited by radial fluctuations in theMaxwell stress. The coherence time of the Maxwell stress is only a few orbits, where as thecorrelation time of the pressure bumps is comparable to the turbulent mixing time-scale, manytens or orbits on scales much greater than one scale height. The particle overdensities contractunder the combined gravity of all the particles and condense into gravitationally bound clustersof rocks and boulders. These planetesimals have masses comparable to the dwarf planet Ceres.I conclude with thoughts on future priorities in the field of planet formation in turbulent discs.
Keywords. diffusion — instabilities — MHD — planetary systems: protoplanetary disks —solar system: formation — turbulence
1. Introduction
Planets form in protoplanetary discs of gas and dust as the dust grains collide andgrow to ever larger bodies (Safronov 1969). An important milestone is the formation ofkm-sized planetesimals. Drag force interaction between particles and gas plays a big rolefor the dynamics of dust particles. This way the collisional evolution of the dust grainsinto planetesimals is intricately connected to the physical state of the gas flow. Themagnetorotational instability renders Keplerian rotation profiles linearly unstable in thepresence of a magnetic field of suitable strength (Balbus & Hawley 1991). The ensuingmagnetorotational turbulence is currently the best candidate for driving protoplanetarydisc accretion. The relatively ease at which self-sustained magnetorotational turbulenceis produced by numerical magnetohydrodynamics codes makes it an excellent test bedfor analysing dust motion and formulating theories of planet formation in a turbulentenvironment.An interesting constraint on the magnetic field present in the solar nebula comes frommeteoritics. Most carbonaceous chondrites have a remanent magnetisation as high as afew Gauss, frozen in as the material cooled past the blocking temperature (Levy & Sonett1978). A quote from the excellent review paper by Levy & Sonett (1978) is particularlyconcise on the origin of such a strong magnetic field:11920 A. Johansen“So far as we can see, there are four major candidates for the origin of the pri-mordial magnetic field which produced the remanence in carbonaceous chondrites.They are:1. Magnetic fields generated in very large meteorite parent bodies2. The interstellar magnetic field compressed to high intensity by the inflowing gas3. A strong solar magnetic field permeating the early solar system4. A hydromagnetic dynamo field produced in the gaseous nebula itself”Levy & Sonett (1978) continue to put forward various physical arguments to rule outpossibility 1 and 2 [the undifferentiated parent bodies of carbonaceous chondrites wereunlikely to harbour a magnetic field, and turbulent diffusion strongly limits the amountof field that can be dragged into the solar nebula (Lubow et al. 1994)]. The magnetic fieldof the wind emanating from the young sun can potentially be strong enough to imprintfields of several G at a few AU from the sun. But the most likely scenario remains thatthe magnetic field was created by the differential rotation and dynamo process in thesolar nebula itself. Simulations of magnetised shear flows indeed show that a weak seedfield can be amplified by the magnetorotational instability to a few percent of the thermalpressure (Brandenburg et al. 1995, Hawley et al. 1996, Sano et al. 2004).In the following sections I briefly review the role of such magnetised turbulence on themotion of dust particles and on the cosmogony of planetesimal formation.
2. Diffusion of small dust grains
The magnetised turbulence in protoplanetary discs moves small dust grains around,preventing them from sedimenting to the mid-plane and transporting dusty materialradially in the disc (Gail 2002, van Boekel et al. 2004). This section describes recent effortsto determine the turbulent diffusion coefficient D t of magnetorotational turbulence.If turbulent transport can be described as a diffusion process, then the evolution ofthe dust particle density ρ d follows the partial differential equation ∂ρ d ∂t = ∇ · (cid:20) D t ρ g ∇ (cid:18) ρ d ρ g (cid:19)(cid:21) . (2.1)Here ρ g is the gas density, its presence signifying that diffusion acts to even out differ-ences in the solids-to-gas ratio ǫ d = ρ d /ρ g . The vertical flux of dust particles containscontributions from the advection (sedimentation at velocity w z ) and the diffusion, F z = w z ρ d − D t ρ g ∂ ( ρ d /ρ g ) ∂z . (2.2)In sedimentation-diffusion equilibrium we have F z = 0. Setting the velocity of the dustparticles its terminal value, w z = − τ f Ω z (where Ω is the Keplerian frequency and τ f isthe friction time of the particles), gives the solution (e.g. Dubrulle et al. 1995) ǫ d ( z ) = ǫ exp[ − z / (2 H ǫ )] (2.3)for the solids-to-gas ratio ǫ d = ρ d /ρ g . The scale height H ǫ follows the expression H ǫ = D t τ f Ω , (2.4) he role of magnetic fields for planetary formation Figure 1.
The dust density at the sides of a simulation box corotating with the disc at anarbitrary distance from the central star. The radial direction points right, the azimuthal directionleft and up, while the vertical direction points directly up. The dust density distribution arisesfrom an equilibrium between sedimentation and turbulent diffusion by the magnetorotationalturbulence. while the solids-to-gas ratio in the mid-plane is given by ǫ = ǫ s(cid:18) HH ǫ (cid:19) + 1 . (2.5)Here H is the pressure scale height of the gas. In the above derivations we have assumed(a) that the friction time is independent of height over the mid-plane and (b) that the dif-fusion coefficient is independent of height over the mid-plane. None of these assumptionsare true in general, but if we stay within a few scale heights of the mid-plane and treatthe diffusion coefficient as a suitably averaged diffusion coefficient, then the expressionsare relatively good approximations.In a real turbulent flow the observed diffusion-sedimentation equilibrium can be usedto measure the turbulent diffusion coefficient of the flow. In figure 1 we show an exam-ple of such a diffusion-sedimentation equilibrium (from Johansen & Klahr 2005) for ashearing box simulation of magnetorotational turbulence. The problem of determiningthe diffusion coefficient is thus reduced to measuring the scale height H ǫ of the dustin figure 1. Using equation 2.4 then directly yields a value of D t . Obviously the diffu-sion coefficient must scale with the overall strength of the turbulence. The interestingquantity to determine is thus the Schmidt number Sc, defined as the turbulent viscositycoefficient relative to the turbulent diffusion coefficient, Sc = ν t /D t . In a Keplerian discthe turbulent viscosity is in turn defined from the Reynolds and Maxwell stresses, ν t = 23 h ρu x u y − µ − B x B y ih ρ i (2.6)22 A. JohansenThe Schmidt number was found by Johansen & Klahr (2005) to be around 1 . .
85 for radial diffusion. This is surprisingly close to unity and a bitmysterious given that the turbulent viscosity is dominated by the magnetic Maxwellstress h− µ − B x B y i . This stress does not directly affect the dust particles. A possibleexplanation is that diffusion is determined by the diagonal entries in the u i u j correlationtensor. These are much higher than the off diagonal Reynolds stress u x u y . Thus the MRIinherently transports a passive scalar (by fluid motion) and the angular momentum (bymagnetic tension) equally well.Different groups have used various independent methods to measure the turbulent dif-fusion coefficient of magnetorotational turbulence. A vertical Schmidt number of aroundunity was measured by Turner et al. (2006), while Fromang & Papaloizou (2006) reporteda value of approximately three. This gives some confidence that the Schmidt numberis well constrained. However, Carballido et al. (2005) found a radial Schmidt number ashigh as ten in relatively strong turbulence. To address the discrepancy between this valueand the much lower value found by Johansen & Klahr (2005), Johansen et al. (2006b)performed simulations of the MRI with various strengths of an imposed, external field,yielding a higher turbulent viscosity than in zero net flux simulations. The Schmidt num-ber was indeed found to decrease with increasing strength of the turbulence. Strongerturbulence (such as in Carballido et al. 2005) is less good at diffusing dust particlesrelative to its stresses. The explanation is that the correlation time of the turbulencedecreases with increasing turbulent energy, and that turbulent structures do not staycoherent long enough to effectively diffuse particles.Large particles partially decouple from the turbulence and are primarily diffused bylarge scale eddies with relatively long correlation times. The experiments by Carballido et al. (2006)indeed showed that the diffusion coefficient falls rapidly for particles above a few metresin size, in good agreement with the analytical derivations of Youdin & Lithwick (2007).
3. Zonal flows
While smaller dust particles are clearly prevented from forming a very thin mid-planelayer by the magnetorotational turbulence, pebbles, rocks and boulders begin to grad-ually decouple from the gas. Accretion discs are radially stratified with a pressure thatdecreases with distance from the star. The pressure gradient acts to reduce the effect ofgravity felt by the gas, and as a result the gas rotates slightly slower than Keplerian.The particles, however, do not react to gas pressure gradients and aim to orbit withthe Keplerian speed. The head wind of the slower rotating gas drains the particles ofangular momentum and they spiral towards the star in a few hundred orbital periods(Weidenschilling 1977).The radial pressure profile of gas in turbulent discs need not be monotoneously falling.The presence of large scale, long-lived pressure bumps leads to concentrations of migrat-ing dust particles into radial bands. Simulations of magnetorotational turbulence in abox gives evidence that such pressure bumps form spontaneously in the turbulent flow(Johansen et al. 2006, Johansen et al. 2009). In figure 2 we plot the gas density and theazimuthal velocity, averaged over the azimuthal and vertical directions, as a function ofradial distance from the centre of the box x and the time t . The gas density exhibitsaxisymmetric column density bumps with amplitude around 5% of the average density.These bumps are surrounded by a sub-Keplerian/super-Keplerian zonal flow, maintainingperfect geostrophic balance with 2 ρ Ωu y ≈ ∂P/∂r .Varying resolution, presence or non-presence of stratification, dissipation parametersand dissipation types, Johansen et al. (2009) find that pressure bumps and zonal flows he role of magnetic fields for planetary formation −1.0 −0.5 0.0 0.5 1.0 x / H t / T o r b ρ /< ρ > 0.95 1.05 −1.0 −0.5 0.0 0.5 1.0 x / H t / T o r b u y / c s −0.05 +0.05 Figure 2.
The gas density (left plot) and the azimuthal velocity (right plot) as a function ofthe radial distance from the centre of the box, H , and the time, t , measured in orbits. There isa perfect − π/ are ubiqituous in shearing box simulations of magnetorotational turbulence, providedthat the simulation box is large enough (more than one scale height in radial extent) andpossibly also that the physical dissipation is high enough. What is then the launchingmechanism for these zonal flows? Large scale fluctuations in the Maxwell stress lead to adifferential transport of momentum. Thus the magnetic field is responsible for separatingthe orbital flow into regions of slightly faster and slightly slower rotation.A model of the excitation of zonal flows and pressure bumps can be obtained from asimplified version of the dynamical equations,0 = 2 Ω ˆ u y − c ρ i k ˆ ρ , (3.1)dˆ u y d t = − Ω ˆ u + ˆ T , (3.2)dˆ ρ d t = − i k ˆ u x − ˆ ρτ mix . (3.3)Here ˆ u x , ˆ u u and ˆ ρ are the amplitudes of the radial and azimuthal velocity and gas densityat the largest radial scale of the simulation, with wavenumber k . The first equationdenotes geostrophic balance, while we have kept the time evolution terms in the twoother equations. Non-linear terms enter through ˆ T , the large scale magnetic tension, andˆ ρ/τ mix , turbulent diffusion of the mass density.We can combine the above equations into a single evolution equation for the density,dˆ ρ d t = 11 + k H (cid:18) ˆ F − ˆ ρ ( t ) τ mix (cid:19) , (3.4)where ˆ F ≡ − k ρ ˆ T /Ω is the forcing term. The prefactor c k ≡ (1+ k H ) − is a pressurecorrection for small-scale modes that both decreases the amplitude of the forcing andincreases the effective damping time. The coherence time-scale of the Maxwell stress (andthus of ˆ F ), τ for , is generally much shorter than the mixing time-scale, τ mix . Thus we needto model equation 3.4 as a stochastic differential equation (see e.g. Youdin & Lithwick24 A. Johansen NL ux ρ Bx2 −BxBy NL uyBy2 Figure 3.
Diagram of how non-linear excitation of the large scale radial magnetic energy leadsto the excitation of zonal flow. Green arrows label positive energy transfer, while red arrows(dashed) denote energy sinks. Non-linear interactions are responsible both for the excitation andfor the balance, the latter through diffusive mixing of the gas density. ρ eq ρ = 2 √ c k τ for τ mix Hk ˆ Tc s . (3.5)The correlation time of the zonal flows is predicted to be equal to the mixing time-scale,in good agreement with the results. The model also predicts that ˆ ρ eq ∝ k − for k H ≫ − B x B y in figure 3) is given, whereas in fact one may go on stepfurther back to B x , which is excited directly by a non-linear term. The Maxwell stressthen increases from the Keplerian stretching of the radial field. The model also predictsthat the magnetic pressure should grow in anti-phase with the thermal pressure. This isindeed also observed.
4. Planetesimal formation
The zonal flows presented in the last section are very efficient at trapping particles.At the outer sub-Keplerian side the particles face a slightly stronger headwind anddrift faster inwards. At the inner super-Keplerian side the particles experience a slightbackwind and move out. The effect of pressure bumps on the migration of rocks andboulders goes at least back to Whipple (1972). It has later received extensive analyti-cal treatment by Klahr & Lin (2001) and by Haghighipour & Boss (2003). The narrow he role of magnetic fields for planetary formation
Figure 4.
The column density of four different particle sizes, before self-gravity has beenswitched on. The particles concentrate at the same locations, but larger particles experiencea higher local column density. box simulations of Hodgson & Brandenburg (1998) found no evidence for long-lived con-centrations of relatively tighly coupled particles in magnetorotational turbulence. How-ever, Johansen et al. (2006a) observed concentrations of marginally coupled dust par-ticles (cm-m sizes), by up to two orders of magnitude higher than the average paricledensity, in high pressure regions occuring in magnetorotational turbulence. In a simula-tion of a (part of a) global disc Fromang & Nelson (2005) reported similar concentrationsin a long-lived vortex structure.The question of how long-lived high pressure structures form and survive in magnetisedturbulence is of general interest. However, their effect on planetesimal formation is noless intriguing. Johansen et al. (2007) expanded earlier models of boulders in turbulenceby considering several particle sizes simultaneously and solving for the self-gravity of theboulders. First the turbulence is allowed to develop for 20 local rotation periods withoutthe gravity of the particles (which is weak anyway). This way a sedimentary mid-planelayer, with a width of a few percent of the gas scale height, forms in equilibrium betweensedimentation and turbulent diffusion. In figure 4 we show the column density of the fourdifferent particle sizes – rocks and boulders with sizes 15 cm, 30 cm, 45 cm, and 60 cm.A weak zonal flow has been sufficient to create bands of very high particle overdensity.An additional instability in the coupled motion of gas and dust has further augmentedthe local overdensities (Goodman & Pindor 2000, Youdin & Goodman 2005, Youdin &Johansen 2007, Johansen & Youdin 2007).As the self-gravity of the disc is activated, the overdense bands contract radially. Uponreaching the local Roche density, a full non-axisymmetric collapse occurs and a fewgravitationally bound clusters of rocks and boulders condense out of the particle layer.The column density of the particles is shown in figure 5.26 A. Johansen
Figure 5.
The column density at ∆ t = 7 T orb after self-gravity is turned on. Four gravitationallybound clusters of rocks and boulders have condensed out of the flow. The most massive cluster(see enlargement) has a mass comparable to the dwarf planet Ceres by the end of the simulation.
5. Conclusions
The presence of magnetic fields in protoplanetary discs is of vital importance for planetformation and for observational properties of protoplanetary discs. Small dust grains aretransported very efficiently by the turbulence. While this counteracts sedimentation tothe mid-plane, and thus prevents the razor thin mid-plane layer of Goldreich & Ward (1973)from forming, the turbulent transport underlies the presence of small dust grains manyscale heights above the disc mid-plane. The presence of crystalline silicates in the coldouter regions of discs (Gail 2002, van Boekel et al. 2004) can likely also be attributedto turbulent diffusion (but see Dullemond et al. 2006 for an alternative view taking intoaccount disc formation history).Larger dust particles – pebbles, rocks, and boulders – slow down or reverse the radialmigration as they encounter variations in the radial pressure gradient. Fluctuations inthe Maxwell stress, with a coherence time of a few orbits, launch axisymmetric zonalflows. These flows in turn go into geostrophic balance with a radial pressure bump.The concentrations of solid particles in such pressure ridges can get high enough for agravitational collapse into planetesimals to occur. However, a satisfactory mechanism forsetting the scale of the pressure bumps is lacking, as the bumps grow to fill the box forall considered box sizes in Johansen et al. (2009). The final size may ultimately be setby global curvature effects (Lyra et al. 2008a).An important problem related to the motion of dust particles in turbulence is theircollision speeds. The relative speed of small particles approaches zero as the particleseparation is decreased. But particles that are only marginally coupled to the turbulenteddies have a significant memory of their trajectories and can collide at non-zero speeds.Carballido et al. (2008) indeed found that the relative speeds of large particles is un- he role of magnetic fields for planetary formation
Acknowledgements
I would like to thank my collaborators Andrej Bicanski, Andrew Youdin, Axel Bran-denburg, Frithjof Brauer, Hubert Klahr, Jeff Oishi, Kees Dullemond, Mordecai-MarkMac Low, Thomas Henning, and Wladimir Lyra.
References
Balbus, S. A., & Hawley, J. F. 1991,
Astrophys. J. , 376, 21Blum, J., & Wurm, G. 2008,
ARA&A , 46, 21Brauer, F., Henning, T., & Dullemond, C. P. 2008,
Astron. Astrophys. , 487, L1Brandenburg, A., Nordlund, ˚A., Stein, R.F., & Torkelsson, U. 1995,
Astrophys. J. , 446, 741Carballido, A., Stone, J. M., & Pringle, J. E. 2005,
Mon. Not. R. Astron. Soc. , 358, 1055Carballido, A., Fromang, S., & Papaloizou, J. 2006,
Mon. Not. R. Astron. Soc. , 373, 1633Carballido, A., Stone, J. M., & Turner, N. J. 2008,
Mon. Not. R. Astron. Soc. , 386, 145Cuzzi, J. N., Dobrovolskis, A. R., & Champney, J. M. 1993,
Icarus , 106, 102Dubrulle, B., Morfill, G., & Sterzik, M. 1995,
Icarus , 114, 237Dullemond, C. P., Apai, D., & Walch, S. 2006,
Astrophys. J. , 640, L67Fromang, S., & Nelson, R. P., 2005,
Mon. Not. R. Astron. Soc. , 364, L81Fromang, S., & Nelson, R. P. 2006,
Astron. Astrophys. , 457, 343Fromang, S., & Papaloizou, J. 2006,
Astron. Astrophys. , 452, 751Gail, H.-P. 2002,
Astron. Astrophys. , 390, 253Gammie, C. F. 1996,
Astrophys. J. , 457, 355Goldreich, P., & Ward, W. R., 1973,
Astrophys. J. , 183, 1051Goodman, J., & Pindor, B. 2000,
Icarus , 148, 537Haghighipour, N., & Boss, A. P., 2003,
Astrophys. J. , 598, 1301Hawley, J. F., Gammie, C. F., & Balbus, S. A. 1996,
Astrophys. J. , 464, 690
28 A. Johansen
Hodgson, L. S., & Brandenburg, A. 1998,
Astron. Astrophys. , 330, 1169Inaba, S., & Barge, P. 2006,
ApJ , 649, 415Johansen, A., & Klahr, H. 2005,
Astrophys. J. , 634, 1353Johansen, A., Klahr, H., Henning, Th. 2006,
Astrophys. J. , 636, 1121Johansen, A., Klahr, H., & Mee, A. J. 2006c,
Mon. Not. R. Astron. Soc. , 370, L71Johansen, A., Oishi, J. S., Low, M.-M. M., Klahr, H., Henning, T., & Youdin, A. 2007,
Nature ,448, 1022Johansen, A., & Youdin, A. 2007,
Astrophys. J. , 662, 627Johansen, A., Youdin, A., & Klahr, H. 2009,
Astrophys. J. , submittedKlahr, H. H., & Lin, D. N. C., 2001,
Astrophys. J. , 554, 1095Kretke, K. A., & Lin, D. N. C. 2007,
Astrophys. J. , 664, L55Levy, E. H., & Sonett, C. P. 1978, in: T. Gehrels (ed.),
IAU Colloqium 52: Protostars and Planets (The University of Arizona Press), p. 516Lubow, S. H., Papaloizou, J. C. B., & Pringle, J. E. 1994,
Mon. Not. R. Astron. Soc. , 267, 235Lyra, W., Johansen, A., Klahr, H., & Piskunov, N. 2008,
Astron. Astrophys. , 479, 883Lyra, W., Johansen, A., Klahr, H., & Piskunov, N. 2008,
Astron. Astrophys. , 491, L41Safronov, V. S. 1969,
Evoliutsiia doplanetnogo oblaka (Nakua)Sano, T., Miyama, S. M., Umebayashi, T., & Nakano, T. 2000,
Astrophys. J. , 543, 486Sano, T., Inutsuka, S.-i., Turner, N. J., & Stone, J. M. 2004,
Astrophys. J. , 605, 321Turner, N. J., Willacy, K., Bryden, G., & Yorke, H. W. 2006,
Astrophys. J. , 639, 1218van Boekel, R., et al. 2004,
Nature , 432, 479Varni`ere, P., & Tagger, M. 2006,
A&A , 446, 13V¨olk, H. J., Morfill, G. E., Roeser, S., & Jones, F. C. 1980,
Astron. Astrophys. , 85, 316Weidenschilling, S. J. 1977,
Mon. Not. R. Astron. Soc. , 180, 57Whipple, F. L. 1972, in: A. Elvius (ed.),
From Plasma to Planet (Wiley Interscience Division),p. 211Wurm, G., Paraskov, G., & Krauss, O. 2005,
Icarus , 178, 253Youdin, A. N., & Goodman, J. 2005,
Astrophys. J. , 620, 459Youdin, A. N., & Johansen, A. 2007,
Astrophys. J. , 662, 613Youdin, A. N., & Lithwick, Y. 2007,
Icarus , 192, 588, 192, 588