arXiv:1007.4159v1 [astro-ph.EP] 23 Jul 2010
eld&Shbr 20) elde l 20)fudthat found (2008) al. et Helled (b) cannot (e.g., (2008); (1997); Boss Jupiter Schubert discs by of & results protoplanetary Helled earlier location prob- to the (a) contrary notable at 2005) are: Rafikov embryos most planet model The giant criticism. the form the of of lems share System. lion’s Solar the the of giants ice re- the nearby make that a could to showed due star photoionisation OB by (2002) envelope gas al. the inside et of moval sedimentation Boss suggested dust (1998) proto-planets. from Boss result gaseous GI, could ac- of cores such context core that the the by In cores predicted model. solid as massive cretion 2009), Gi- contain Nettelmann System 1997). self- & Solar (Fortney Boss massive the 1974; in a planets Bodenheimer in ant (e.g., born disc condensations co gravitating by gaseous directly e.g., of “big”, traction form planets giant disc. that the suggests en- from gas gas massive of a accretion masses, by Earth up 10 reaches builds about core velope of solid the mass When solid critical larger masses. a ever planet form (planetesi- terrestrial to to objects stick up and km-sized collide 2005; of make then Goodman which to 1990) mals), & 2007) Youdin Wetherill al. (see et (e.g., somehow poorly Johansen a up step then joining pro- necessary is grains the There but in grains. into understood dust grows microscopic in chap- disc form First, and toplanetary planets scenario. that (2008), bottom-up stipulates Lin (2010)) a & Armitage Ida in (1996); 4-6 Safronov ters see al. (e.g., et model Pollack (CA) oppo- (1969); accretion from core starts for- The action planet ends. making site for planet the theories which competing in mation two currently are There INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. Received Nayakshin Sergei planet giant of downsizing tidal embryos by planets of Formation c eateto hsc srnm,Uiest fLeicester of University Astronomy, E-mail: & Physics of Department 08RAS 2008 fteetomdl,teG steoeta received that one the is GI the models, two these Of model (GI) Instability Gravitational the contrast, In [email protected] 000 – 20)Pitd2 a 08(NL (MN 2018 May 29 Printed (2008) 1–5 , ihgsgatpae ihasldcr mre rmteenvelope. s the in from later, emerges happens core it solid If a left. with is t planet core while giant quickly, planet gas occurs terrestrial inne rich disruption a the tidal (iii), pee reach If stage irradiation clump. clumps in stellar the gas to of due the envelope evaporation when and poor shear (v) tidal core; star, solid the the onto collapses i)i h oi oebcmsmr asv than massive more earlier terrest becomes as form core to (iii) solid clumps star; the gas parent the if the inside (iv) c sediments to clumps dust the in (1998), (ii) closer Boss masses; Jupiter migrate few simultaneously a and of masses with clumps gaseous ieyugpoolntr icfamnsat t fragments migratin of disc consists of formation protoplanetary downsizing planet young tidal for sive by scheme made proposed are The embryos. planets that hypothesise We ABSTRACT n- ecse,L17H UK 7RH, LE1 Leicester, , s ya re fmgiueta hs tde yHelled by studied Jupiter those 10-20 about than ( below masses masses magnitude with GEs of co-authors. and order an by hsadesn on a drse on b automaticall formation. (b) core point solid addresses of (a) sites point excellent addressing Thus be to MNRAS) to ted oapa nMRS h ugsin fBs (1998); assuming Boss GEs in of at sedimentation formed suggestions dust are they on the that (2002) MNRAS) al. et in Boss appear to do planets massive let- 1980). as then Tremaine star inwards, and & radially parent (Goldreich 2005), Giant migrate (Rafikov the of allowed them from is formation ting the far process – of the hereafter) ideas version where (GE modern 1990-ies Embryos can more the above planet with upgrades raised model points one the GI if of addressed All itself. be outlived long has like planets terrestrial form to way and no Earth. systems, be Jupiter- the to planetary of seems abundance extrasolar there the in (d) explain observed to planets rare like too are stars OB ro r u ob irpe,te a led contain already may they em- disrupted, the be time to the due by are Thus, that values. time bryos find nor- migration parameter we which the However, plausible than star, distance. for shorter the is AU time of few sedimentation field a grain GE tidal at the the occurs that by mally assume disrupted we internal equal, be to are can due two shrink Hill these can their When they star, cooling. note parent than We the faster to decreases disc. closer radius protoplanetary migrate embryos massive as young that realistic a in utsdmnaini o lwapoest il observed yield mass of to embryos process giant a in slow cores solid too is sedimentation dust ehv eetyrvstd(ae ,Nykhn2010a, Nayakshin I, (paper revisited recently have We criticism this that show to is Letter our of goal The eo eetmt h irto ieo h embryos the of time migration the estimate we Below M A T E J tl l v2.2) file style X r on pprI,Nykhn21b submit- 2010b, Nayakshin II, (paper found are ) R ∼ ∼ eea est udeso Uon AU of hundreds to tens several 0M 10 R p ∼ ⊕ asv a atmosphere gas massive a , 0 U uhGsaecooler are GEs Such AU. 100 eeses i mas- a (i) steps: hese ffteotrmetal- outer the off l ilms oi cores; solid mass rial ∼ < ae(v,ametal a (iv), tage uie’ as (c) mass; Jupiter’s o n contract, and ool esse sstill is system he e Ufrom AU few r in planet giant g ugse by suggested y. 2 S. Nayakshin
4 1/2 massive solid cores. Depending on the properties of the GE 1/2 10 yrs Remb (t) = 0.8 AU κ∗ . (4) at disruption, removal of the outer metal poor gas shell re- t veals either a terrestrial planet or a metal-rich giant planet. We therefore argue that such a “tidal downsizing” scheme may explain planets of all types without recourse to plan- 2.2 Dust growth and sedimentation etesimals (Safronov 1969; Wetherill 1990). We echo the approach of Boss (1998) here; the resulting Our hypothesis is a hybrid scenario: giant planet forma- grain growth time scale in a constant density embryo, for a tion starts with dust sedimentation inside a self-gravitating grain with initial size a0 to reach the final size a is gas clump (Boss 1998), as in the GI model, but continues as in the CA model with accretion of a massive metal-rich gas 3cs a tgr,0 = ln , (5) envelope from within the GE. Additionally, solid cores left πfgρ0GR0 a0 by the GE disruption in the protoplanetary disc may grow 3 where ρ0 = 3Memb/4πR , cs and fg 0.01 are the initial further like in the CA model – by accreting smaller solid emb ≈ embryo mean density, the sound speed, and the grain frac- debris from the disc. Giant planets may accrete more gas tion by mass, respectively. In paper I we improved on this and also migrate radially, as in the “standard theory”. model by allowing the embryo to contract, in which case the growth time is a geometric combination of the cooling and 3/7 4/7 the initial grain growth time, tgr = t0 tgr,0: 2 GIANT PLANET EMBRYO MODEL 4/7 3 −2/7 −4/7 9/21 ln(a/a0) tgr = 3 10 yr m1 f−2 k∗ , (6) 2.1 Birth and contraction × 20
Massive young gas discs are gravitationally unstable and where f−2 = fg/0.01. The vaporisation time, tvap, is the fragment on clumps when the disc cooling time is no longer time it takes for the embryo to heat up to the vaporisation than a few times the local dynamical time, 1/Ω (e.g., temperature of Tvap 1400 K. This is trivially obtained 3 1/2 ≈ Gammie 2001), where Ω = (GM∗/Rp) is the Keple- by solving the equation T (tvap) = Tvap (using eq. 1), and a rian angular frequency at radius Rp from the star of mass good approximation is > M∗. This limits the fragmentation region to Rp 30 100 2 − 4 Tvap −2 AU (e.g., Rafikov 2005; Stamatellos & Whitworth∼ 2008; tvap = 1.5 10 years m1 κ∗ , (7) × 1400 K Meru & Bate 2010). At the point of marginal gravitational stability the density in the disc midplane is about the tidal Grains sediment to the centre of the core when tgr < tvap, 3 and vaporise before they could sediment if tgr >tvap. density at that point, ρt = M∗/(2πRp). Thus, when the disc just fragments, the disc cooling time is about the free-fall The mass of refractory material (e.g., silicates) that can timescale for the material in the disc, tff 1/√Gρ. As the be used to build a massive core is expected to be about 1/3 ∼ ∼ clumps contract further, their cooling time becomes longer of the total heavy elements mass, or about 20 Earth masses than their free-fall times. The initial state of a GE is thus for Memb = 10MJ (Boss et al. 2002). However, due to en- that of a polytropic clump (Cameron et al. 1982), similar ergy release associated with the solid core accretion, strong to the “first cores” studied in the context of star formation convective motions near the core may develop. The grains (Larson 1969; Masunaga & Inutsuka 2000). For simplicity also melt at too high core accretion rates. Thus the final we assume that GEs evolve at constant mass after their for- mass of the core is usually much smaller than the maximum mation. condensible mass (paper II). Introducing dust opacity as a power-law in temperature and assuming that the embryo is optically thick, we obtained 2.3 Radial migration of embryos an approximate analytic solution for the radiative cooling of the giant embryo in paper I1. For example, for κ(T ) = Gravitational torques between the disc and massive κ0(T/10K), we have planets are significant (e.g., Goldreich & Tremaine 1980;
1/2 1/2 Tanaka et al. 2002). While GEs are smaller than their Hill’s t 4/3 −4/9 t T (t)= T0 1 + 2 = 146 m1 κ∗ 1 + 2 , (1) radii (see below), we assume that they migrate at similar h t0 i h t0 i rates. Under this assumption, embryos migrate via type I 3 where T0 is the temperature of the embryo at formation, regime if Memb 6 Mt = 2M∗(H/R) and via type II regime with m1 = Memb/10MJ being the mass of the gas clump in if Memb > Mt, where Mt is the transition mass (Bate et al. units of 10MJ , k∗ = κ0/0.01, and t0 is the initial cooling 2003). For a self-gravitating, Q = 1 disc, the type II migra- time of the clump: tion time is 2/3 1/9 R2 2 t0 = 380 years m κ∗ . (2) −1 −1 p 4 −1 3/2 0.2Rp 1 tII (Rp)= α Ω = 4 10 yrs α R , (8) H2 × −1 2 H Note that the cooling time of the embryo increases with where α−1 = α/0.1 is the disc viscosity parameter and R2 = time, so that tcool(t) t when t t0. At a constant mass ∝ ≫ Rp/100AU. Type I migration is faster: the embryo radius decreases with time as −1 M∗ H 3 3/2 H −1 4/9 −1/2 R0 κ∗ t tI (Rp) = Ω = 3 10 yrs R2 m1 , (9) Memb Rp × 0.2Rp Remb = 1/2 = 6.0 AU 1/3 1 + 2 . (3) [1 + 2t/t0] m t0 1 h i We can improve these estimates by building a simple disc At large t, independently of Memb, all GEs contract as model that would specify how disc properties change with
c 2008 RAS, MNRAS 000, 1–5 Tidal downsizing hypothesis 3
R. We define the stellar mass doubling time scale as tdb = 1000 10000 M∗/M˙ where M˙ is the mass accretion rate on the star, as- sumed to be constant in time and radius inside the disc. By 100 the order of magnitude, tdb should be comparable to the free fall time of the host gaseous cloud from which the star forms, 5 e.g., 10 yrs for a 1 M⊙ cloud at the typical interstellar ∼ temperature of 10 K (e.g., Larson 1969). 10 ** * Using the standard Shakura & Sunyaev (1973) accre- [AU] p
tion disc formalism, we first find H(R) and Md(R) in the R innermost disc. We then find the radius beyond which the Toomre (1964) parameter falls below unity. In the outer self-gravitating region of the disc we follow the treatment 1 Core formation of Goodman (2003) for a disc with no internal energy 1000 10000 sources and enforce Q (H/R) (M∗/Md) = 1. Finally, ≈ 2 for a constant accretion rate disc, M˙ = 3παH ΩΣd, where 2 10 Σd Md(R)/(πR ) is the column depth of the disc, we can ≈ RH solve for [AU] H H 1 1/3 R = . (10) emb R h 3αΩtdb i and R The accretion rate disc model built in this way is schematic 1 but correctly predicts the expected location of the self- emb R gravitational region at Rp > few tens of AU to 100 AU. Using this disc model we are now able to calculate the mi-
1000 10000 gration rate of a GE at an arbitrary Rp for a given tdb. We follow the fit to numerical results by Bate et al. (2003) to 1000 smoothly join the type I and type II migration regimes. The fastest migration then occurs for embryo mass Memb = Mt.
2.4 Tidal disruption ** * The Hill’s radius of the embryo is (for M∗ = 1 M⊙) 100 1/3 Memb 1/3 RH = Rp = 0.15 Rpm1 . (11) h 3M∗ i Embryo temperature [K] In general, radial migration accelerates as the GE moves in, 1000 10000 whereas the contraction of the envelope slows down with time [years] time (cf. equation 4). Therefore, as planets migrate inwards, there is a point where the embryo’s radius Remb becomes Figure 1. Evolution of GEs of mass 3, 6 and 10 Jupiter masses comparable to the GE Hill’s radius. The tidal field from the (black solid, red dotted and blue dashed, respectively) calculated parent star at this point starts to peel off the outer layers as described in §2. Upper panel: Radial position of the embryos of the embryo. The tidal radius, Rt, is defined as the radius versus time. The GEs start off at R = 100 AU. The coloured < where Remb = ηtRH (ηt 1). Using the type II migration asterisks and text “core formation” show tgr – the time when a estimate as an example, we∼ insert eq. 8 into eq. 4, and find solid core should form inside the embryo, with the colour matched to that of the curve. Middle: Radius of the first core (power-law − H − −1 1/2 3/4 1/2 1/3 like curves), and the tidal radius at the current location of the Rt = 2.8 AU ηt α−1 R2 κ∗ m1 . (12) 0.2R first curve. The first core is assumed to be completely disrupted In Appendix we show that irradiation by the parent star can when the two sets of curves meet, i.e., when the tidal radius drops R heat up the outer layers of the envelope and even disrupt below emb. Lower: Temperature of the first cores as a function the whole GE. The effect appears less important than tidal of time, with core formation times marked as above. disruption for Solar type stars, but may become dominant for higher mass stars. The upper panel shows radial migration of the embryos. Note that initially all migrate at about the same rate (in the type II regime), but then the least massive embryo overtakes 3 AN ILLUSTRATIVE EXAMPLE the other two. The least massive one (black solid curves) Figure 1 shows the results of our giant embryo model de- migrates the fastest as it happens to fall in the minimum of scribed in 2 for embryos of 3, 6 and 10 Jupiter masses, the migration time curve, between type I and type II regimes § shown with different colours. The parameters used for the (see fig. 11 in Bate et al. 2003). The asterisks and the text 5 figure are tdb = 10 yrs, α = 0.1, κ0 = 0.01, M∗ = 0.5 M⊙ “core formation” mark the time t = tgr, when the solid core (as the star is assumed to be in the midst of its growth, is assumed to form in the centre of the GEs (cf. eq. 6). rather than close to being completely assembled), Solar met- The middle panel shows the embryo size, Remb, and alicity, and the starting radial position is Rp = 100 AU. the respective Hill’s radius, for the three cases considered.
c 2008 RAS, MNRAS 000, 1–5 4 S. Nayakshin
> We assume that when the two sets of curves intersect, the in Armitage 2010), which yields Mcr 3 M⊕ (see paper II). embryo is disrupted, leaving only the core, which migrates The maximum value of Mcr can be∼ obtained by employ- at a negligible rate during the calculation. This is why the ing the classical CA model results – the radiative zero so- radial position curves, Rp(t), approach a constant value in lution (Stevenson 1982; Ikoma et al. 2000), and this yields < the upper panel at late times. Mcr 20 M⊕. The lower panel shows the respective temperature evo- We∼ would thus posit that, if the tidal disruption of the lution of the embryos (cf. eq. 1). The least massive embryo GE occurs when the core mass exceeds Mcr, the outer metal- forms the core last, as it is the least dense of the three. How- poor low density layers of the GE are peeled off, but the ever, it is also the coolest one. Due to this, the least massive inner high density solid core plus the bound gas atmosphere embryos may lock most of their “metal” content into grains remains. The disruption leftover must be a metal rich (com- and actually yield heavier solid cores (see paper II). On the pared to the parent star) giant planet, as are the giant plan- other hand, the least massive embryo is disrupted sooner (in ets in the Solar System. accord with eq. 12), at Rp 6 AU. ≈ 4.1.3 Giant planets without solid cores? 4 THE TIDAL DOWNSIZING HYPOTHESIS The GE may go through the second wave of collapse when the temperature reaches about 2000-2500 K (Larson 1969; 4.1 Planets: leftovers of disrupted GE ? Bodenheimer 1974; Masunaga & Inutsuka 2000) which can compact the gas component to densities as high as g cm−3. We found that for plausible conditions, the GE initial cool- ∼ ing time, the dust growth time, the grain vaporisation time The time to reach that temperature is at least a few times and the GE migration time are related by the vaporisation time (cf. eq. 7). Therefore, if t0 tgr −5/21 growth timescale behaves as tgr fg (eq. 6), whereas Irradiation can also disrupt the GE when the irradiation ∝ 2 1/4 tvap fg . A higher fg thus gives the GE more time to build temperature, Tirr =(Lirr/4πR σ) is comparable to the ∝ emb a solid core. Further, Meru & Bate (2010) find that lower mean temperature (equation 1) inside the GE, e.g., Tirr = < dust opacities discs fragment closer to the star, reducing ηevT (t), with ηev 1. For one of the fixed planet location migration time (cf. eq. 8). Low metalicity GEs are likely to runs Cameron et al.∼ (1982) found ηev 0.5 (cf. their Fig. 4 1/2 ∼ vaporise their grains before they make a solid core, and they 2). As Rev =(L∗/4πσT0 ) (t0/2tII ), we have also migrate inward too quickly to their ultimate demise. 2 2 −3/2 H 0.5 −1/3 Rev 1 AU k∗α−1R2 m1 (17) Challenges and open questions. “Tidal downsizing” is ≈ 0.2Rp ηev a complex hypothesis that may turn out to not work for at For fiducial parameters, tidal disruption of GEs occurs ear- least the following reasons. (1) giant planet embryos need to lier (at larger radii) than the irradiative evaporation. Fur- be isolated or slowly accreting. If they pile up mass quickly thermore, as noted by the referee, the giant embryo may be by accretion from the disc, they become too hot to support shadowed by the disc if the disc scale-height, H, is larger grain sedimentation (paper I). (2) We assumed that GEs mi- than the GE size, Remb. Thus we expect tidal disruption of grate as planets do. This is probably a good approximation the GEs to be more frequent. Note however different param- when the radius of the embryo Remb RH, but much more eter dependencies in eq. 12 and 17. In general both processes ≪ work is needed to confirm this. (3) We have seen that the may be important in removing GE envelopes, with irradia- timescales of the important processes (see 4.1) do depend tion becoming more important for higher mass stars. § strongly on dust opacity and other parameters. Real proto- planetary discs may be such that grain growth inside the GEs is precluded or the embryos are disrupted too quickly. REFERENCES Armitage P. J., 2010, Astrophysics of Planet Formation Bate M. R., Bonnell I. A., Bromm V., 2003, MNRAS, 339, 6 CONCLUSIONS 577 Bodenheimer P., 1974, Icarus, 23, 319 The tidal downsizing hypothesis combines some of the planet Boley A. C., Hayfield T., Mayer L., Durisen R. H., 2010, building processes from both the CA and the GI models, pre- Icarus, 207, 509 serving the strengths but not the weaknesses of these theo- Boss A. P., 1997, Science, 276, 1836 ries. Open questions remain, but on the balance we feel that Boss A. P., 1998, ApJ, 503, 923 the hypothesis deserves a further detailed consideration by Boss A. P., Wetherill G. W., Haghighipour N., 2002, Icarus, the planet formation community. We note that, having sub- 156, 291 mitted this paper and paper I, we discovered an independent Cameron A. G. W., Decampli W. M., Bodenheimer P., suggestion on the possibility of terrestrial core formation by 1982, Icarus, 49, 298 tidally disrupted giant embryos by Boley et al. (2010). Clarke C. J., Lodato G., 2009, MNRAS, 398, L6 Fischer D. A., Valenti J., 2005, ApJ, 622, 1102 Fortney J. J., Nettelmann N., 2009, Space Science Reviews, 7 ACKNOWLEDGMENTS 115–+ Gammie C. F., 2001, ApJ, 553, 174 The author acknowledges illuminating discussions with and Goldreich P., Tremaine S., 1980, ApJ, 241, 425 comments on the draft by Richard Alexander, S.-H. Cha, Goodman J., 2003, MNRAS, 339, 937 Phil Armitage, Adam Burrows and Roman Rafikov. Cathie Helled R., Podolak M., Kovetz A., 2008, Icarus, 195, 863 Clarke, the referee, is thanked for suggestions that signifi- Helled R., Schubert G., 2008, Icarus, 198, 156 cantly improved the clarity of the paper. Theoretical astro- Ida S., Lin D. N. C., 2008, ApJ, 685, 584 physics research at the University of Leicester is supported Ikoma M., Nakazawa K., Emori H., 2000, ApJ, 537, 1013 by a STFC Rolling grant. Johansen A., Oishi J. S., Low M., Klahr H., Henning T., Youdin A., 2007, Nature, 448, 1022 Larson R. B., 1969, MNRAS, 145, 271 Masunaga H., Inutsuka S.-i., 2000, ApJ, 531, 350 APPENDIX: IRRADIATIVE EVAPORATION Meru F., Bate M. R., 2010, ArXiv e-prints Irradiation of the GE by the star can affect the former Mizuno H., 1980, Progress of Theoretical Physics, 64, 544 strongly (Cameron et al. 1982). First we estimate the radius Pollack J. B., Hubickyj O., Bodenheimer P., Lissauer J. J., Rlock below which the temperature of the surface layers of Podolak M., Greenzweig Y., 1996, Icarus, 124, 62 the GE is set by the irradiation rather than the internal Rafikov R. R., 2005, ApJL, 621, L69 radiation. This is found by equating the stellar radiation lu- Safronov V. S., 1969, Evoliutsiia doplanetnogo oblaka. 2 2 minosity incident on the GE, Lirr = L∗πRemb/(4πRp), to Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 the isolated GE radiative luminosity, Lfc(t). Since Lfc(t) Stamatellos D., Whitworth A. P., 2008, A&A, 480, 879 −1 −1/2 ∝ T (t) (1+2t/t0) for the GE (paper I), the result for Stevenson D. J., 1982, P&SS, 30, 755 ∝ Rlock at t = tII and L∗ = L⊙ is Tanaka H., Takeuchi T., Ward W. R., 2002, ApJ, 565, 1257 Toomre A., 1964, ApJ, 139, 1217 1/2 −1 3/4 1/4 −3/8 H Wetherill G. W., 1990, Annual Review of Earth and Plan- Rlock 12 AU m1 k∗ α−1 R2 . (16) ≈ 0.2Rp etary Sciences, 18, 205 c 2008 RAS, MNRAS 000, 1–5 6 S. Nayakshin Youdin A. N., Goodman J., 2005, ApJ, 620, 459 c 2008 RAS, MNRAS 000, 1–5