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arXiv:1901.06448v2 [astro-ph.EP] 3 Mar 2019 omto cnroi urnl bevdi eltm ythe by time real in ( observed actu high-resolution currently is “bottom-u latest new is object entirely scenario This central formation processes. protosate the accretion and before by primor- protoplanets formed Jovian long forming the respectively, of and lites, solar capable the nebulae pre- of dial we models 2019a,b), isothermal Kazanas & sented (Christodoulou work previous In Introduction 1. cm g nosaltlwrrttoa upr gis self-gravity dis against ’ support also, rotational region; ( disk lower flat-density Jupiter’s lot outer an a for enjoys need no is ik r infiatydi significantly are disks n uie,rsetvl)adterszs(.0G ess1 versus Gm cm Uranus g (0.60 (55.6 for sizes densities km, central their and 368 respectively) and Gm, versus respectively) km Jupiter, (27.6 and lengths scale radial their rela evolution. in of form years could of millions over orbiting safety the tive a which similarities in find minima pri tential to Uranus’ and of same nebula model the di Jupiter’s best-fit our to apply compare nebula we to sa mordial 2018; al. is major work, et six goal al. Pérez its et Our this formed Pineda that lites. disk In 2018; 2018; primordial Uranus’ 2019). to al. al. model et al. al. et Long et et Huang Favre al. Marel 2018; et der 2018; Isella van 2018; al. et al. Kudo 2018; et al. Dullemond 2018; 2017, et al. al. Harsono et et 2018; Clarke Guzmán al. et 2018; al. Lee 2018; et 2018; Zhang al. 2017; et al. 2018; Avenhaus et Keppler Ruane 2018; Partnership (ALMA 2018; al. 2016; et telescope Macías al. ALMA 2018; et Andrews the by 2015; disks tostellar iis rns oei mle yaotafco f2( 2 of factor a about by smaller is core Uranus’ tities: m,terda est rfiei hloe ( shallower is profile density radial the Gm), ff oe fapoolntr ikfrigi-iutemjrUra major the in-situ forming disk protoplanetary a of Model rne ewe h w ik hthse rvttoa po- gravitational hosted that disks two the between erences swsepce,tetomdlnbleaevr di very are nebulae model two the expected, was As − Keywords. su nua eoiyi bu . ie hto uie’ oe(a of core Jupiter’s density of central that the times hand, 2.3 about other is the velocity On angular Jovian ( Gm). smaller the 0.60 lot like (only a look disk and is all state self-gravity at against of not support equation does centrifugal the disk disk, This pro the nebula. Uranus’ of Uranian of length model scale density radial oscillatory the isothermal an fit We 2019 5, March 3 2 1 3 epciey.I diint tutrldi structural to addition In respectively). , ffi -al demos.kazanas@.gov E-mail: NASA Low Massachusetts [email protected] of E-mail: Univ. Sciences, Mathematical of Dept. oelCne o pc cec n ehooy University Technology, and Science Space for Center Lowell inl lwt urne t ogtr tblt gis s against stability long-term its guarantee to slow ciently / SC aoaoyfrHg-nryAtohsc,Cd 6,G 663, Code Astrophysics, High-Energy for Laboratory GSFC, β 0 lnt n aelts yaia vlto n stability and evolution dynamical satellites: and planets ≈ 5 × 10 ∼ ff - U bevtoso aypro- many of observations AU) 1-5 rn nterrmiigpyia quan- physical remaining their in erent − 3 versus aeltsbfr h lnti formed is planet the before satellites iirsM Christodoulou M. Dimitris β 0 ≈ 3 × 10 k − ≈ − 2 respectively). , − ff 3 ) n there and 1), rne,the erences, ess0.31 versus R ff β 1 rn in erent 0 ≈ oainpro f30da poe o68d.Yt h rotatio the Yet, d). 6.8 to opposed as d 3.0 of period rotation = ABSTRACT than ally l-rvt nue ntblte o iloso years. of millions for instabilities induced elf-gravity h eta est ftepioda a,adterotationa the and gas, primordial the of density central the 0 tel- 0 nd . p” ikta emdldpeiul.Isrtto aaee tha parameter rotation Its previously. modeled we that disk olntr ikt h rsn-a ao aeltsadwe and satellites major present-day the to disk toplanetary . 00) si h ailsaelnt ol 76k)adtesi the and km) 27.6 (only length scale radial the is as 00507), l- 2 k 1 1 - - h opc rna oei ihrb atro 8 n t co its and 180 of factor a by higher is core Uranian compact the l,Lwl,M,084 USA. 01854, MA, Lowell, ell, fMsahstsLwl,Lwl,M,084 USA. 01854, MA, Lowell, Lowell, Massachusetts of , 2 n eotee Kazanas Demosthenes and n ecmaetebs-trslst uie’ extended Uranus Jupiter’s of to moons results major follow best-fit the 2 what the to nebula compare In model we here. our and 2 descriptions § the in in the repeating apply detail and we of in disk, need Jupiter’s described no for been is (2019b) have Kazanas nebula resulti & the gaseous Christodoulou and the equation of Lane-Emden model isothermal the of tions .Pyia oe fUau’Poolntr Disk Protoplanetary Uranus’ of Model Physical 2. h rtpaeaycr fe thdbe itdsvrl by severely tilted impact. been giant a had early it because aro after formed core is was pla disk protoplanetary the This accretion the to the inclination degrees. that means zero 98 this nearly and at of equator knocke orbit tilt moons was inner axial protoplanet the of current the its before actu to not was over but maj planet formed; its the fully before that long ally signify formed disk were compact moons equatorial Uranus’ in speeds rotation etdb aaise l 19) hsmto (M method This imple (1998). as al. et algorithm op- Lagarias simplex the by Nelder-Mead and mented 1999) the al. used et Shampine timization 1997; Reichelt & (Shampine rvstebs tsbtnily h edfricuigCor including for need inclusion The substantially. its fit because best small ) the proves innermost and , Mir The (, , satellites Uranus. other , the of with along moons used was the of axes jor nume stable extremely slow. somewhat it albeit makes cally, which procedure search its in fminsearch rfie eepromdwt h M densi the oscillatory with produce performed that were integrations profiles model numerical disk The Uranian Best-Fit 2.1. paesadstlie:frainpoolntr disks formation—protoplanetary satellites: and —planets n§3 esmaieaddsusorresults. our discuss and summarize we 3, § In . h nltc(nrni)adnmrcl(siltr)solu (oscillatory) numerical and (intrinsic) analytic The h xrml ihgsdniisadteml di mild the and densities gas high extremely The nFg ,w hwtebs piie tt h semima- the to fit optimized best the show we 1, Fig. In enet D271 USA. 20771, MD reenbelt, osntueaynmrclo nltclgradients analytical or numerical any use not does ) 3 atlab fteds is disk the of n ode15s tt fthe of state l measures t determine eo the of ze atlab nian ae1o 4 of 1 Page integrator ff re’s erential routine Model very a anda, delia net’s und im- ng ri- or re ty s, ll d - - - Christodoulou & Kazanas

moon Oberon (Rmax ≈ 0.60 Gm). The next outer density peak lies at a distance of 0.95Gm around which no moonis known.In fact, the disk of Uranus must have been really small (< 1 Gm in radial extent) because the next outer , , -6 has a semimajor axis of 4.3 Gm. C -8 P M A -10 U T 2.2. Physical parameters from the best-fit model O -12 R R2 Using the scale length of the disk 0 and the definition 0 = -14 c2 G 0/(4π ρ0), we write the equation of state for the Uranian cir- -16 cumplanetary gas as

-18 c2 0 GR2 × 6 5 −1 −2 = 4π 0 = 6.39 10 cm g s , (1) -20 ρ0 -7 -6 -5 -4 -3 -2 -1 0 where c0 and ρ0 are the local sound speed and the local density in the inner disk, respectively, and G is the gravitational constant. T c2 RT For an isothermal gas at , 0 = /µ, where µ is the mean molecular weight and R is the universal gas constant. Fig. 1. Equilibrium density profile for the midplane of Uranus’ pri- Hence, eq. (1) can be rewritten as mordial protoplanetary disk that formed its largest moons. The inner- T most small moon Cordelia was also retained because it improves the fit. −3 ρ0 = 13.0 g cm , (2) (Key: C:Cordelia, P:Puck, M:, A:Ariel, U:Umbriel, T:Titania, µ ! O:Oberon.) The best-fit parameters are k = −0.96, and β0 = 0.00507 R − (or, equivalently, 1 = 0.0967 Gm). The radial scale length of the disk where T and µ are measured in degrees Kelvin and g mol 1, R is only 0 = 27.6 km. The Cauchy solution (solid line) has been fitted respectively. to the present-day so that its density maxima (dots) For the coldest gas with T ≥ 10 K and µ = 2.34 g mol−1 correspond to the observed semimajor axes of the orbits of the moons (molecular hydrogen and neutral helium with fractional abun- (open circles). The density maximum corresponding to the location of X Y Titania was scaled to a distance of RT = 0.4358 Gm. The mean relative dances = 0.70 and = 0.28 by , respectively), we find error of the fit is 7.5%, affirming that this simple equilibrium model pro- that duces a good match to the observed data points. The intrinsic solution −3 (dashed line) and the nonrotating analytical solution (dash-dotted line) ρ0 ≥ 55.6 gcm . (3) are also shown for reference. This extremely high value implies that the conditions for proto- satellite formation were already in place during the early isother- mal phase (Tohline 2002) of the Uranian nebula. stems from the structure of the oscillatory solutions of the Lane- Using the above characteristic density ρ0 of the inner disk in Emden equation (Lane 1869; Emden 1907) with rotation: the so- the definition of ΩJ ≡ 2πGρ0, we determine the Jeans fre- lutions track closely the nonrotating solution at small radii and quency of the disk: p they turn around to approach the intrinsic solution (creating os- −3 −1 cillations) only after they cross below the intrinsic core density ΩJ = 4.8 × 10 rad s . (4) 2 value of τ = β0 (Jeans 1914). Even for extremely low values of the parameter β , these solutions create density maxima deep Then, using the model’s value β0 = 0.00507 in the definition of 0 ≡ inside the core region, so there is need for at least one moon β0 Ω0/ΩJ, we determine the angular velocity of the uniformly- R ≤ residing well inside the core and close to the center R = 0. rotating core ( 1 0.0967 Gm), viz. We have effectively used only two free parameters (k and −5 −1 Ω0 = 2.5 × 10 rad s . (5) β0) to fit the current orbits of the seven satellites, and the best- fit model is of good quality (mean relative error of 7.5%, all of For reference, this value of Ω0 for the core of the Uranian nebula which is coming fromthe positions of Umbriel and Oberon).The corresponds to an of P0 = 3.0 d. This value is inner core parameter R1 is strongly correlated to β0, and we did close to the present-day orbital period of Ariel (2.5 d), but it is not use an outer flat-density region beyond the radius R2 because not near the orbital period of the largest moon Titania (8.7 d). the disk is too small in radial extent. This is a deviation from what we found for the and We find the following physical parameters from the best-fit for Jupiter. Nevertheless, the large outer moons of Uranus are all model: k = −0.96 and β0 = 0.00507 (equivalently, R1 = 0.0967 comparable in mass and size, so our previous finding remains Gm, just beyond the orbit of Puck and closer to the orbit of the valid: the angular velocity of the core of the primordial nebula minor moon ). The radial scale of the model was determined is comparable to the present-day angular velocities of the largest by fitting the density peak that correspondsto the orbit of Titania regular satellites. to its distance of 0.4358Gm, and the scale length of the disk then turns out to be R0 = 27.6 km. The best-fit model is certainly sta- ble to nonaxisymmetric self-gravitating instabilities because of 2.3. Comparison between the models of Uranus and Jupiter the extremely low value of β0 (the critical value for the onset of We show a comparison between the physical parameters of the dynamical instabilities is β∗ ≃ 0.50; Christodoulou et al. 1995). best-fit models of Uranus and Jupiter in Table 1. Obviously,these The model disk extends out to 1 Gm (ln R = 0 in Fig. 1), two protoplanetary disks are very different in most of their phys- but its validity ends around the distance of the outermost major ical properties. The disk of Uranus is a lot smaller (Rmax), more

Page 2 of 4 Formation of the major Uranian satellites

Table 1. Comparison of the protoplanetary disks of Jupiter and Uranus

Property Property Jupiter’s Uranus’ Name Symbol(Unit) Model2 Best-FitModel Density power-law index k −1.4 −0.96 Rotational parameter β0 0.0295 0.00507 Inner core radius R1 (Gm) 0.220 0.0967 Outer flat-density radius R2 (Gm) 5.37 ··· Scale length R0 (km) 368 27.6 c2 5 −1 −2 × 9 × 6 Equation of state 0/ρ0 (cm g s ) 1.14 10 6.39 10 −3 Minimum core density for T = 10 K, µ = 2.34 ρ0 (g cm ) 0.31 55.6 −1 Isothermal sound speed for T = 10 K, µ = 2.34 c0 (m s ) 188 188 −1 −4 −3 Jeans gravitational frequency ΩJ (rad s ) 3.6 × 10 4.8 × 10 −1 −5 −5 Core angular velocity Ω0 (rad s ) 1.1 × 10 2.5 × 10 Core P0 (d) 6.8 3.0 Maximum disk size Rmax (Gm) 12 0.60

compact (R0), and denser (ρ0) by a factor of 180. In addition, the up” scenario in which regular satellites form first, followed by disk of Uranus is just as cold (assuming that T = 10 K), a lot their planets, and then by the central star. heavier (ΩJ), and its core is rotating (Ω0) more than twice as fast Our modeling efforts have produced very different models (still, this is a slow rotation with a period P0 of only 3.0 d). for Uranus’ and Jupiter’s disk. This supports the optimization The power-law index of the Uranian nebular model is k ≈−1 procedure that led to such different results. This model appears (surface density Σ ∝ R−1), unlike the Jovian nebula and the so- to be capable of navigating between varied physical conditions lar nebula (k ≈ −1.5, Σ ∝ R−1.5). This range of values of k and of finding the best-fit model in each different case. The re- has been observed in studies of young circumstellar disks in the sults for the two gas giants so far indicate that at least two minor pre-ALMA era (Andrews & Williams 2007; Hung et al. 2010; planets form inside the uniform core of the model, but the major Lee et al. 2018, and references within). planets form along the intrinsic density gradient characterized by Although the disk of Uranus is small, it is still very heavy the power-law index (k − 1). However, there is still a large region and hosts high densities of gas, thus also of ices and planetesi- in the inner core whereno moonsform;this regionis whererings mals. As we found for Jupiter’s disk, these conditions support a form around the gas giants. These rings must also have formed at “bottom-up” hierarchical formation in which protosatellites are density maxima, and they may have to be taken into account in seeded early inside such nebular disks and long before their pro- future modeling of Saturn and Neptune. Fitting the same model toplanets are fully formed; these compact moon systems com- to these gas giants encounters more difficulties than the models plete their formation in < 0.1Myr(Harsono et al. 2018)andlong of Jupiter and Uranus. We are in the process of investigating the before the central stars become fully formed (Greaves & Rice protoplanetary disks of Saturn and Neptune as well. 2010). References 3. Summary ans Discussion ALMA Partnership, Brogan, C. L., Pérez, L. M., Hunter, T. R., et al. 2015, ApJ, We have constructed isothermal differentially-rotating proto- 808, L3 Andrews, S. M., & Williams, J. P. 2007, ApJ, 659, 705 planetary models of the Uranian nebula, the primordial disk in Andrews, S. N., Wilner, D. J., Zhu, Z., et al. 2016, ApJ, 820, L40 which the regular moons were formed (§ 2). The best-fit model Avenhaus, H., Quanz, S. P., Garufi, A., et al. 2018, ApJ, 863, 44 is shown in Fig. 1 and its physical parameters are listed in Ta- Christodoulou, D. M., & Kazanas, D. 2019a, Part 1, arXiv: 1901.02593 (Solar ble 1. In the optimization, we retained also the smaller moons Nebula) Christodoulou, D. M., & Kazanas, D. 2019b, Part 3, arXiv: 1901.05131 (Jupiter) Cordelia and Puck in order to fit the nearest two density maxima Christodoulou, D. M., Shlosman, I., & Tohline, J. E. 1995, ApJ, 443, 551 to the center that form inside the uniform core of the disk. This Clarke, C. J., Tazzari, M., Juhasz, A., et al. 2018, ApJ, 866, L6 allowed us to find a better model than when major moons are Dullemond, C. P., Birnstiel, T., Huang, J., et al. 2018, DSHARP VI, ApJ, 869, assumed to have formed inside the core. The remaining mean L46 Emden, R. 1907, Gaskugeln, Leipzig, B. G. Teubner relative error of 7.5% was due to inaccuracies in the positions of Favre, C., Fedele, D., Maud, L., et al. 2018, ApJ, arXiv:1812.04062 only two moons, Umbriel and Oberon. Greaves, J. S., & Rice, W. K. M. 2010, MNRAS, 407, 1981 We have compared this model to the best-fit model of Jupiter Guzmán, V. V., Huang, J., Andrews, S. M., et al. 2018, DSHARP VIII, ApJ, 869, (Christodoulou & Kazanas 2019b) (§ 2.3). Uranus’ disk has a lot L48 Harsono, D., Bjerkeli, P., van der Wiel, M. H. D., et al. 2018, Nature , less centrifugal support which makes it extremely stable against 2, 646 dynamical nonaxisymmetric instabilities. Furthermore, this disk Huang, J., Andrews, S. N., Dullemond, C. P., et al. 2018, DSHARP II, ApJ, 869, is much smaller, heavier, and denser than the disk of Jupiter. De- L42 spite these values, Uranus’ inner core rotates about twice as fast, Hung, C.-L., Lai, S.-P., & Yan, C.-H. 2010, ApJ, 710, 207 Isella, A., Huang, J., Andrews, S. N., et al. 2018, DSHARP IX, ApJ, 869, L49 although its temperature (as measured by the isothermal sound Jeans, J. H. 1914, Phil. Trans. Royal Soc. London, Series A, 213, 457 speed; Table 1) is assumed to be the same as that of the Jovian Keppler, M., Benisty, M., M¨uller, A., et al. 2018, A&A, 617, A44 disk. Kudo, T., Hashimoto, J., Muto, T., et al. 2018, ApJ, 868, L5 Both of these models appear to be stable and long-lived, so Lagarias, J. C., Reeds, J. A., Wright, M. H., & Wright, P. E. 1998, SIAM Journal of Optimization, 9, 112 their regular moons can form early in the evolution of each neb- Lane, L. J. H. 1869-70, Amer. J. Sci. Arts, 4, 57 ula and long before the protoplanetsmanage to pull their gaseous Lee, C.-F., Li, Z.-Y., Ho, P. T. P., et al. 2017, ApJ, 843, 27 envelopes on to their solid cores. The results support a “bottom- Lee, C.-F., Li, Z.-Y., Hirano, N., et al. 2018, ApJ, 863, 94

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Long, F., Pinilla, P., Herczeg, G. J., et al. 2018, ApJ, 869, 17 Macías, E., Espaillat, C. C., Ribas, Á, Schwarz, K. R., et al. 2018, ApJ, 865, 37 Pérez, L. M., Benisty, M., Andrews, S. N., et al. 2018, DSHARP X, ApJ, 869, L50 Pineda, J. E., Szulágyi, J., Quanz, S. P., van Dishoeck, E. F., et al. 2018, ApJ, arXiv:1811.10365 Ruane, G., Mawet, D., Kastner, J., et al. 2017, AJ, 154, 73 Shampine, L.F., & Reichelt, M. W. 1997, SIAM Journal on Scientific Computing 18, 1 Shampine, L.F., Reichelt, M. W., & Kierzenka, J. A. 1999, SIAM Review 41, 538 Tohline, J. E. 2002, ARA&A, 40, 349 van der Marel, N., Dong, R., di Francesco, J., et al. 2019, ApJ, arXiv:1901.03680 Zhang, S., Zhu, Z., Huang, J., et al. 2018, DSHARP VII, ApJ, 869, L47

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