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Lecture 7 Protoplanetary disks Part III Lecture Universität Heidelberg WS 11/12 Dr. C. Mordasini Based partially on script of Prof. W. Benz Mentor Prof. T. Henning Lecture 7 overview 1. Viscous accretion disks continued 1.1 Analytical solution: steady state 1.2 Analytical solution: time dependent 1.3 Temperature structure 1.4 Energy budget and SEDs 2. Disk dispersion by photoevaporation 2.1 Internal photoevaporation 2.2 External photoevaporation 3. Central magnetospheric cavity 1.1 Analytical solution: steady state Steady state We recall the master equations for the angular momentum transport: T=torque j=specific ang. momentum In general, there is no analytical solution to the equation as it is non-linear. Lyden-Bell & Pringle 1974 have worked out a number of analytical solutions to the master equation under simplifying assumptions. We study three of them with increasing complexity. Steady state solution with zero torque inner boundary condition A special solution is the simple case that the mass flux is constant as a function of radius, i.e. In that case, the master equation can simply be integrated on both sides: To determine the integration constant T0, we assume that the star is rotating slower than with breakup speed. This means for the angular frequency of the star’s rotation at the surface Steady state II Then, there must exist a boundary layer at Rc where the disk's Ω changes from the (faster) Keplerian value to the slower stellar value. i.e. where Ω T=0 The torque is proportional to the gradient of the angular velocity, therefore the torque vanishes at the place where Ω turns over at Rc. Therefore, Ω★ R★ at Rc R where the second approximation holds for a small viscosity, i.e. a negligible thickness of the boundary layer. The specific angular momentum is simply It should be noted that is the angular velocity of the star grazing orbit, and not the real angular velocity of the stellar surface (which is as said smaller than this). We therefore find for the steady state solution Steady state III With that we can calculate all quantities torque T, mass flux, surface density and radial velocity of the gas as Note Given a viscosity, this equation defines the steady-state surface density profile for a disk with a constant accretion rate. Away from the boundaries, Σ(r) ∝ ν−1. Assuming an alpha viscosity, we can make this more explicit: The link with the scale height means that here, the vertical structure of the disk comes in, or in other words, the temperature structure, which is a priori unknown. Steady state IV The last equations mean that . Suppose we know that Tmid is a given power-law: Shakura & Sunyaev model Then, away from the inner boundary, Examples: For the passively irradiated disk, we have Tmid~ r"1/ 2 ! " ~ r#1 ! Dullemond Note however that this is not self-consistent because we have to take into account viscous heating, but it illustrates how disk surface density and temperature are linked. Shakura & Sunyaev model Examples: Steady state V To get a surface density profile like for the MMSN, we must have a constant disk temperature (unlikely!) Tmid~ r"0 " ~ r#3 / 2 ! ! Dullemond In the last figures, the effect of the inner boundary was neglected. Taking this into account, we find The radius of the sun is today about 0.0046 AU. Initially, it was larger by a factor of a few. 1.2 Analytical solution: time dependent Time dependent solutions Time dependent solution for a point mass without central couple We now look at time dependent solutions in a fixed outer gravitational field. The change of the gravitational field due to accretion onto the central object is thus neglected. Rewriting the master equation, using the continuity equation in cylindrical coordinates and assuming that ν(r) varies as a powerlaw of r, yields the following partial differential equation: where for the central point mass: Solutions T(j,t) or equivalently T(r,t) to this differential equation can be resolved in modes in which the variation of T with the time t is proportional to General solutions are then found as a superposition of such solutions, where their specific form is given by the boundary conditions. For the case of T(h=0,t=0)=0 (no central couple), and the initial condition Time dependent solutions II It is found after some algebra that it is convenient to use in this case the dimensionless time tν =1 corresponds thus to the initial time. The solution is then given as Radius of maximum viscous couple (or radius of velocity reversal) From the flux or vr it can be seen that for inwards inside the corresponding radius the velocity is outwards, and This position is also the position of maximum viscous couple (torque) as we can easily determine from finding the maximum of T(r,tν). Time dependent solutions III Time dependent solution for a point mass with central couple We can use the result from the steady state case to construct a time dependent solution with central couple. The equations in the last section indicate that near the center, the time dependent solution has the same form as the steady state solution without a central couple: Thus, we assume that if we now want to include the central couple with the star, we can replace this expression with the one for the steady state with central couple, leaving the rest unchanged. For a physically usable solution, we must determine the constants a and C. The constant a comes from the initial profile of the disk, and is related to the initial size of the disk R1 (the place where the surface density is e-0.5 the value at the center in the solution without central couple) as The constant C is found from the total disk mass: We find: Time dependent solutions IV Putting these expression together we finally find: (r>>R ) ★ The dimensionless time can now be written as where is the viscous timescale measured at R1. We also have: Example Parameters Disk mass [Msun] 0.020 0.018 0.016 0.014 0.012 0.010 0.008 0 2 4 6 8 10 Stellar accretion rate [Msun/yr] t [τ] Notes •Disk mass and accretion rate gradually decrease. •The process is fastest at the beginning. •The numerical values are similar to observed ones (for the chosen value of e.g. the viscosity). 1 � 10�7 5 � 10�8 2 � 10�8 1 � 10�8 5 � 10�9 2 � 10�9 1 � 10�9 0 2 4 6 t [τ] 8 10 Surface density evolution t=0, 0.1, 0.3 and 1 τ 200 Σ [g/cm2] Σ [g/cm2] 100 50 20 10 a[ AU ] τ] [ t 0 10 20 30 a[AU] Note -disk is getting larger in time. -surface density is decreasing in time. 40 50 Gas velocity evolution t=0, 0.1, 0.3 and 1 τ t [τ] 400 vr [cm/s] vr [cm/s] 200 0 �200 �400 1 5 10 50 100 500 a[AU] a[AU] 50 RMVC [AU] Note •inner part of disk is accreting. •outer part is decreting. •the point of velocity reversal at RMVC moves itself outwards as the time increases, overtaking more and more material (it moves outwards faster than the gas itself). Gas initially not much outside RMVC first flows outwards, and inwards somewhat later on. •in the end, an infinitesimal amount of mass is at infinity, containing the angular momentum. 40 30 20 10 0 2 4 6 t [τ] 8 10 Gas flux [Msun/yr] Gas flux evolution t [τ] a[AU] 1.5 � 10�7 We see a wave like feature traveling outwards, decreasing in amplitude. Gas flux [Msun/yr] 1. � 10�7 5. � 10�8 0 �5. � 10�8 �1. � 10�7 �1.5 � 10�7 1 2 5 10 20 a[AU] 50 100 200 1.3 Temperature structure Temperature To get the temperature of the disk, we consider the heating by viscous dissipation and the cooling at the disk surface by radiation. Let us compute the vertically integrated rate of viscous dissipation [erg/ (s cm2)] which is given as the vertical integral over stress times rate of strain T = 2πr � ∞ rσrϕ dz −∞ For the steady state disk, we have far from the inner boundary for the value of the torque so we find (quasi Keplerian) This heat must be transported away. Since for a thin disk the z gradient dominates the radial gradient, the heat equation is well approximated by as the vertical heat transport inside the disk can happen through several mechanism. Temperature II At the two disk surfaces, we must finally have cooling as a black body: For the thin steady accretion disk far from the inner boundary this gives Midplane temperature Teff is the temperature at the surface. If the heat transport is radiative, and the disk is optically thick, we can estimate the midplane temperature which depends on the optical depth. The optical depth is defined as (κRoss is the Rosseland mean opacity) The basic equation for vertical radiative flux F(z) in the diffusion approximation is: For simplicity, we assume that all energy is liberated at z=0 (in the midplane). Then, F(z) is constant with z and given as . Assuming that κRoss is also constant with z, we can integrate the differential equation. Temperature III For the second line we have used that for τ ≫ 1 almost all of the disk gas lies below the photosphere. For large τ, Tmid4 ≫ Teff4, and the last line simplifies to We finally find for the midplane temperature due to viscous dissipation This equations provides the requested link of temperature (vert. structure) and surface density (radial structure). For full solutions, we first need an expression for the Rosseland mean opacity. We can then express the midplane temperature as a function of the surface density. In the radial structure equations we can then eliminate the temperature (e.g. in the viscosity) in exchange for the surface density. We then have an equation entirely in terms of the surface density which we can solve. Temperature IV In general, such solutions can only be calculated numerically, as e.g. the opacity is not a simple analytical function. - α-disk - viscous heating only (Alibert et al 2004) - opacity transitions lead to different regimes - temperature set by accretion rate - this also sets the position of the ice-line, which is important for giant planet formation. - α-disk - includes both viscous dissipation and stellar irradiation (Fouchet et al 2011) - and photoevaporation (Mordasini et al. 2011) - alpha 7 x 10-3 - temporal evolution of the central temperature (one line all 20‘000 years). - as the disk mass decreases, T approaches the irradiation only profile. 1.4 Energy budget and SEDs Energy budget We can compute the amount of energy dissipated in the disk between the inner and the outer boundary: where to obtain the last equality we assumed a Keplerian disk. Note that this energy is equal to half the total gravitational energy, the outer half has gone into orbital kinetic energy. This kinetic energy must be dissipated in the boundary layer near the star. In other words, the equivalent of the entire energy dissipated in the disk must be dissipated in the boundary layer. Since this region is narrow, the energy dissipated heats up the layer to high temperatures so that short wavelength radiation is generally associated with the boundary layer (UV excess). "F" Wien region Spectral energy distribution SED SEDs provide important RayleighJeans region observational information on the structure of protoplanetary disks. Each annulus of the disk radiates as a black body. The maximum of the radiation of a black body at temperature T is found according ! to Wiens law at Dullemond The observed SED is the superposition of the contributions of all annuli in the multi-color region. At short wavelength, there is the Wien region, while at long wavelength, there is the Rayleigh-Jeans regions, SED II A disk with a power law temperature distribution that extends over a large enough distance tends to radiate a power law spectral energy distribution. To see this, we realize that each optically thick annulus 2 π r dr contributes mostly with photons from the peak of the local black body distribution. From Wien’s law, we have for the steady state disk Thus, the emergent radiation will have the form and therefore For the general case, one can show that for we have Multi-color blackbody disk SED Rayleigh-Jeans region: !F! !(4q-2)/q Slope is as Planck function: "F" # " 3 Multi-color region: !3 " SED III SED of SED of accr % % 3 Remember: Remember: While this would provide a well observable signature of a viscous accretion disk, itTis = ' Teff M =˙ '$ eff & 8"# flat& 8 unfortunately not the only model that produces such a SED. It was pointed out that a simple disk reprocessing the light received from a central ball will haveAccording the same SED. This because According toisour derived to our derived SED ru 2 in such a case the light drops with the usual 1/r compounded with the geometrical factor which is proportional to R/r where R is the radius of the central object. " F" # " ! ! Does this fitofSEDs of Her Does this fit SEDs Herbig Ae/ Observed SEDs from T-Tauri stars ˙ = 2 "10 M /yr need M ˙ need M = 2 "10 M /yr ! HD104 ! #7 sun #7 sun ! HD104237 ! Adams et al. 1990 Dullemond et al. While some SED have a shape compatible with the purely viscous accretion disk model, others have a flatter SED. Flat SED imply a temperature dependance with radius: Teff ∝ r-1/2 as opposed to the steeper dependance derived for the purely viscous disks with Teff ∝ r-3/4. Distinguishing passive irradiation from active viscous accretion is difficult from the SED. 2. Photoevaporation Photoevaporation Self-similar solution of a viscous disk: surface density and accretion rate decline as powerlaws at late times: => Smooth transition between disk and diskless states expected. Not observed: YSO between CTTS and WTTS (so called “transition objects”) are rare. From observed statistics, one infers that the time scale for clearing the disk is short, only ~105 yrs. An additional mechanism besides viscosity is needed to allow such a behavior. An observational indication to the mechanism that additionally could drive disk dispersal was provided by HST observations of low mass stars exposed to the strong ionizing flux produced by neighboring massive stars (O-stars) in the core of the Orion Nebula’s Trapezium cluster. The images show drop shaped nebulae surrounding young stars with protoplanetary disks, which are interpreted as the signature of photoevaporation and escape of disk gas. Observations of photoevaporation Basic mechanism •Irradiation of disk surface creates a thin surface layer of hot gas with T>Teff. •If its temperature/thermal velocity exceeds the local escape velocity, the surface layer gets unbound and evaporates, i.e. a thermal wind is launched taking away disk gas. Basic types depending upon origin of the disk-irradiating flux 1) External photoevaporation: the radiation originates from other stars than the host star, usually nearby massive stars in young clusters that have prodigious ultraviolet luminosities as it is the cae in Orion. 2) Internal photoevaporation: for disks around stars forming in groups or small clusters (e.g. Taurus) the influence of external photoevaporation is small (no nearby massive stars), and mass is instead driven internally, by the radiation from the central star itself. Relevant radiation types 1) FUV: Far-ultraviolet radiation. 6 eV < hν < 13.6 eV. Dissociates hydrogen molecules but does not ionize hydrogen atoms. 2) EUV: Extreme-ultraviolet. 13.6 eV < hν < 100 eV. Ionizes hydrogen. 3) X-rays. Defined by convention as photons having hν > 0.1 keV. Internal photoevaporation Armitage 2010 For internal photoevaporation, irradiation by all three types of radiation can play a role. Here we focus on EUV driven evaporation as modeled by Clarke, Gendrin & Sottomayor 2001. The host star EUV flux leads to a temperature of the ionized disk surface layer of The mean molecular weight of the ionized gas μII is about 0.6. This leads to a speed of sound in the gas The mean thermal velocity is similar to the sound speed. . Internal photoevaporation II Gas with this temperature is unbound from the star beyond the so called gravitational radius For 1 Msun star, the gravitational radius is at about 7 AU. Numerical simulations show that some mass is already lost at somewhat smaller distances beyond . The resulting evaporation can then be estimated as where n0(r) is the number density of ionized atoms at the base of the heated layer. The latter is found in detailed photo-hydrodynamical simulations as (1) (2) (3) In equation (2), kH is a constant, and Φ41 is the stellar output of ionizing photons in units of 1041 photons per second. The origin of this radiation comes from both the background photospheric flux, and the hard radiation from the stellar boundary accretion layer. The latter establishes a link with the viscous evolution of the disk. Internal photoevaporation III Note the square root dependence of the number density on Φ41 in (2). It comes from the fact that the flow is ionization/recombination limited. This means that most of the photons are expended maintaing the ionization level in the flow, and only a small fraction is used to create the mass flux from the interface between the neutral/ionized interface at the disk surface. The equilibrium between the photoionisation rate of neutral atoms with a number density nbase and the radiative recombination rate of ionized atoms with density n0 at the surface means [1/(cm3 s)] where is the photoionisation cross section of H for UV radiation [cm2] while αrec is the radiative recombination coefficient for hydrogen ions [cm3/s]. Therefore, . Equation (3) finally means that the flow is mostly concentrated near Rwind. For typical parameters (kH=5.7x107 , Φ41=1, βII=0.69 i.e. Rwind=5 AU for 1Msun; Hollenbach et al. 1994) this gives a constant mass loss of about 3 x 10-10 Msun/yr. As we shall see below, this becomes important towards end of disk evolution, and causes the formation of a hole/gap in the disk at about Rwind. Internal photoevaporation IV Alexander et al. 2005 Illustration of a photohydrodynamic simulation, showing a wind flowing away from the inner disc due to internal photoevaporation. Density is plotted as a color scale, with the ionization front denoted by a dashed line and the computational boundaries denoted by solid lines. Velocity vectors are plotted at regular intervals. The plot shows a state late in the disk evolution, when the inner disk (inside of the gap) has already disappeared. The disk is now directly irradiated by the star and cleared in an inside-out way. External photoevaporation Very close to massive stars (<0.03 pc) external EUV photoevaporation is important. At more typical (larger) distances, FUV radiation from the massive stars dominates, as it is less strongly attenuated by the background gas in the stellar cluster. This external FUV causes the formation of a hot neutral layer on the disk’s surface (Matsuyama, Johnstone & Hartmann 2003). The mass loss rate is now estimated as: The gravitational radius is now of order Rg,I≈ 140 AU. Mass loss occurs outside this radius. Here, rmax is the outer disk radius, β=0.5 and the total mass loss rate is a parameter which is depends e.g. on the distance from the massive star providing the heating radiation. This process takes away mass outside about 70 AU. It is important from the beginning by reducing the external disk radius and by mass loss there. Combined model Surface density [g/cm2] 10000 ’fort.14’ βRg,Iu 2:3 Rwind Combined viscous evolution and internal & external photoevaporation. 1000 100 α parameter of 7 x 10-3. 10 Viscous dissipation and stellar irradiated included for the temperature structure. 1 0.1 1-2 x MMSN disk (Mdisk(t=0)=0.024 M⊙ 0.01 0.001 Mean Ṁwind,ext=7 x10-9 Msun/yr. 0.1 Mordasini et al. 2011 1 10 Semimajor axis [AU] 100 1000 Resulting disk lifetime τ=2.01 Myrs. A red line shows the gas surface density all 20000 yrs. The arrows show the direction of the mass flux at the end state. Combined model II Characteristic phases 1. The initial exterior radius specified by the initial conditions gets quickly reduced by external photoevaporation to a radius where mass removal due to photoevaporation, and the viscous spreading of the disk are in a quasi-equilibrium as described. In the specific case, the radius decreases from initially about 200 AU to ∼ 100 AU. In the inner part, the disk very quickly evolves from the initial profile towards equilibrium. 2. In the second, dominant phase a quasi self-similar evolution of the disk occurs. The inner part of the disk (r ≤ 10 AU) is in near steady state (i.e. the mass accretion rate is nearly constant as a function of radius), and the slope of gas surface density γ is approximately -0.9 (but varying between -0.4≤ and -1.5 due to opacity transitions). The outer radius is slowly decreasing, here from about 100 AU down to 60 AU. For more massive disk, this equilibrium radius is further out. 3. Once the viscous gas accretion rate in the inner disk becomes comparable to the mass loss rate due to internal photoevaporation (happens when the surface density has fallen to about 0.01 to 0.1 g/cm2 at ∼10 AU), a gap opens somewhat outside of Rwind. The evolution of the disk now speeds up which correspond to the so called “two-timescale” behavior (Clarke et al. 2001). 4. Quickly afterwards, the total disk mass has fallen to 10−5 M⊙, where the calculation is stopped. The effect of the direct radiation field which would clear the disk quickly from inside out once the gap has opened is not included in the simulation shown here. 3. Central magnetospheric cavity Central part of the disk The central part of the protoplanetary disk (a<0.1 AU) has a complicated structure as temperatures are high (dust evaporation, ionization), stellar irradiation is strong and stellar magnetic fields start to become important. We here focus on the aspect of magnetospheric accretion (cf. Bouvier et al. 2006). The final accretion of gas onto the star (and ejection via jets) takes place in the compact magnetized region around the central star. The inner disk edge is interacting with the star’s magnetosphere, leading simultaneously to magnetically channeled accretion flows and to high velocity winds and outflows. The magnetic star-disk interaction is thought to have strong implications for the angular momentum evolution of the central system, the inner structure of the disk, and possibly for halting the migration of young planets close to the stellar surface (Hot Jupiters). Surface fields of order of 1-3 kG have been derived from Zeeman broadening measurements of CTTS photospheric lines. The plot shows measured stellar magnetic field strength of T Tauri stars in Orion, Taurus/Auriga and the TW association, as a function of time. Yang et al. 2011 Basic structure Assuming that T Tauri magnetospheres are predominantly dipolar on the large scale, we show below that the inner accretion disk is expected to be truncated by the magnetosphere at a distance of a few stellar radii above the stellar surface for typical mass accretion rates of 10−9 to 10−7 M⊙/yr. Disk material is then channeled from the disk inner edge out of the plane onto the star along the magnetic field lines, giving rise to magnetospheric accretion columns. The free falling material in the funnel flow eventually hits the stellar surface where an accretion shock develops near the magnetic poles of the star. Size of the cavity The size of the magnetospheric cavity can be estimated by assuming a spherical accretion flow onto the star with free fall velocity, and a dipolar magnetic field. The ram pressure of the accreting material is This will at some point be offset by the magnetic pressure for a sufficiently strong stellar field. Where these two pressures are equal, if the accreting matter is sufficiently ionized, its motion will start to be controlled by the stellar field. This point is usually referred to as the truncation radius RT. If we consider the case of spherical accretion, we have For a dipolar stellar magnetic field we also have Size of the cavity II Equating the two pressures and solving for R gives For typical values this corresponds to where B3 is the stellar field strength in kG, is the mass accretion rate in units of 10−8 M⊙ /yr, M0.5 is the stellar mass in units of 0.5 M⊙, and R2 is the stellar radius in units of 2 R⊙. Notes -the magnetospheric cavity has thus the size of about 7 stellar radii, or roughly 0.05 AU. -a strong magnetic field makes a larger cavity. -a low gas accretion rate makes a large cavity. This could indicate that the cavity expands towards the end of the disk lifetime. Size of the cavity III In the case of disk accretion (instead of spherical accretion), the numerical coefficient above is different, but the scaling with the stellar and accretion parameters remains identical. In an accretion disk, the radial motion due to accretion is relatively low while the Keplerian velocity due to the orbital motion is only a factor of 21/2 lower than the free-fall velocity, i.e. similar. The low radial velocity of the disk means that the disk densities are much higher than in the spherical case, so that the disk ram pressure is higher than the ram pressure due to spherical free-fall accretion. As a result, the truncation radius will move closer to the star. In this regard, the equation for RT gives an upper limit for the truncation radius. Therefore, models typically use a inner disk radius of where k≤1. In reality, CTTS have more complex magnetic field topologies, which need not to be aligned with the rotation axis. In this case, numerical simulations are necessary to get the structure of the cavity. A slice of the funnel stream obtained in 3D simulations for an inclined dipole (Θ = 15◦). The contour lines show density levels,The thick lines depict magnetic field lines (Romanova et al., 2004a). examine the Lindblad torque for each n because of the exchange of the angular momentum between vertical modes with di†erent n. Only the Lindblad torque summed over n is well deÐned. If the scale height increases with r, this e†ect -decreases ! the di†erential torques, as shown in Table 2. AsStrong a B Weak B result of an increase in h with r, the !<<0 vertical averaging is at 0.1 AU no cavity more e†ective for outer THREE-DIMENSIONAL Lindblad resonances than forcavity the DISK-PLANET No. 2, 2002 I !≈0.9 inner ones. This makes outer Lindblad resonances weak and decreases the di†erential torques. is more appr smaller thanThe thatthree in two-dimensional disks of the asymmetric e†ects on the because total corotation the pressure ine†ectiveness of !the gradient. torque, (a) pressure , !(d) , and !(k) , are listed in Table 3. Since CT,n torque CT,n is obtained CT,n from the solution around the corotation To see how corotation, !(k)5. DISCUSSION is well deÐned for each n. For n \ 0, as the two-dim CT,n readily seen from equation (55), only !(a) has a nonzero setting n \ k CT,0toThe 5.1. Orbital Migration of Planets due the contribuvalue, and it is given by ! (a) \ [2! (curv)/3. CT,0 CT,0 MMSN, irrad., Σ0 ∝ a-0.9, Distance [AU] equation (21 Distance [AU] Disk-Planet Interaction tion from n [ 0 is small in all e†ects as well as in the curva,alpha=7x10-3 coordinate x ture e†ect. As stated in ° 1, the radial migration of a planet due to the As a zero order approximation, canresults, take the of the cavity intodi†eraccount by Lindblad forcing the res From the we above we effect can calculate the total interaction is distance, an and important process intorque planet surface disk-planet density to fall to zero at some as total shown on the left. ential Lindblad torque the corotation in obtain formation. general In thethree-dimensional previous section obtained combining the total disks.we Furthermore, wewith obtain the total disk leads to As we will see later, the, Lindblad gravitational oftorques, a planet the through gaseous torque, ! the exerted and on interaction acorotation three-dimensional disk total netplanet, torque on the itdisk, ! , for three-dimensional disks torquesthe acting on the causing to Since migrate radially. Fortorque low mass planets in so called disk-planet interaction. the total on a total 1.1 1.2 two-dimensional disks as type I migration, theand torque is: Σ Gas surface density [g/cm2] Gas surface density [g/cm2] Implication for planetary migration A A B B planet has the opposite sign, the radial migration speed of M r ) 2 where b4 the planet, r5 , !is given by p p p (3D) \ (1.364 ] 0.541a) p r 4 ) 2 , p total p p p p M c ] 2)/ /() c! p,m dL ~1 the Lin Torque(69) causes change ofnear planet’s angular p Tanaka M rtotal ) ,2 \ [2r r5 \ L0 rbation excited by the et al. 2002 , (68)L , i.e. radial migration. !p (2D)p\dr (1.160 ] 2.828a) p p Lp p p r4 )2momentum equation (72 ation is normalized in p p p total M c p malized surface Thus, thedensity inner cavity with α<<0 is ap strong stoppingc mechanism for low mass planets would not ch where momentum of Lthree-is ones in type II. where we substituted k \ 3/2, torque, d \ the 3/2but ] planet anot for for themassive migrating in typethe I as itangular causes a strong outward introduce a p dimensional case and d \ a for the two-dimensional case. M (GM r )1@2. Since ! is usually positive, the migration A B p Questions?