On the Terminal Rotation Rates of Giant Planets

A standard approach to modeling astrophysical disks is to assume that their temperature, T, and mid-plane density,¹ ρ, profiles take the form of power laws in orbital radius (Armitage 2010). Here, we follow this convention and adopt the following functional forms:

$$\begin{eqnarray}T & = & \min \left[{T}_{\max },{T}_{0}{\left(\displaystyle \frac{{R}_{{\rm{H}}}}{r}\right)}^{3/5}\right]\\ \rho & = & {\rho }_{0}{\left(\displaystyle \frac{{R}_{{\rm{H}}}}{r}\right)}^{5/2},\end{eqnarray} \tag{ 1 }$$

where ${R}_{{\rm{H}}}$ = a (M/3M_⋆)^1/3 = 0.36 au is the Hill radius, T_max = 1500 K, T₀ = 250 K, and ρ₀ = 1.5 × 10⁻⁹ kg m⁻³ (Figure 1). These simple relations provide a good match to the simulated disk of Szulágyi et al. (2016) and Szulágyi (2017), where a Jupiter-mass planet is held at a constant surface temperature of 1500 K. We note, however, that unlike the quoted hydrodynamical simulations, our broken power-law temperature profile does not rise above T_max, yielding an isothermal disk interior to r < 0.05 ${R}_{{\rm{H}}}$ (as we will see below, our analysis is insensitive to the exact functional form of T at sufficiently high temperatures).

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Although conditions attained within a disk parameterized by Equations (1) are insufficient to generate Hydrogen or Helium ions in any appreciable fraction, temperatures in excess of T ≳ 1000 K are high enough to thermally ionize alkali metals, which are expected to be present in trace amounts within the nebula (Armitage 2010). Accordingly, assuming that the disk material is an ideal gas of solar composition, we have computed the ionization levels of Na, K, Li, Rb, Fe, Cs, and Ca, employing the Saha equation:

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{i}^{+}\,{n}_{e}}{{n}_{i}-{n}_{i}^{+}}={\left(\displaystyle \frac{{m}_{e}{k}_{{\rm{b}}}T}{2\pi {{\hslash }}^{2}}\right)}^{3/2}\exp (-{I}_{i}/{k}_{{\rm{b}}}\,T),\end{eqnarray} \tag{ 2 }$$

where ${n}_{i}^{+}$ and n_i are positive ion and total number densities of constituent i, n_e is the total electron number density, m_e is the electron mass, k_b and ℏ are Boltzmann's and Plank's constants, and I_i is the ionization potential of constituent i. The appropriate elemental abundances and ionization potentials were adopted from Lodders (1999) and Cox (2000), respectively. The electrical conductivity within the circumplanetary disk was then computed using the standard expression

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{{n}_{e}}{{n}_{n}}\displaystyle \frac{{e}^{2}}{{m}_{e}\,{{ \mathcal A }}_{c}}\sqrt{\displaystyle \frac{\pi \,{m}_{e}}{8\,{k}_{{\rm{b}}}\,T}},\end{eqnarray} \tag{ 3 }$$

where e is the electron charge, n_n is the number density of neutral particles, and ${{ \mathcal A }}_{c}$ is the number density weighted scattering cross-section of Hydrogen and Helium.

The electrical conductivity profile of the disk is shown in Figure 1. Within the inner ∼5% of the Hill radius, σ ranges from ∼0.005–0.3 S/m—a value 1–2 orders of magnitude smaller than the conductivity of salty water. Crucially, however, the conductivity itself is not the primary quantity of interest. Instead, magnetic induction within the disk is characterized by the magnetic Reynold's number (Moffatt 1978):

$$\begin{eqnarray}&&{{ \mathcal R }}_{m}=\displaystyle \frac{{v}_{\mathrm{kep}}\,h}{\eta }={\mu }_{0}\,\sigma \,r\,\sqrt{\displaystyle \frac{{k}_{{\rm{b}}}\,T}{\mu }},\end{eqnarray} \tag{ 4 }$$

where η = 1/μ₀ σ is the magnetic diffusivity, μ₀ is the permeability of free space and μ = 2.4 m_H is the mean molecular weight of the gas. As shown in Figure 1, the magnetic Reynold's number exceeds unity by orders of magnitude out to orbital distances comprising a substantial fraction of the Hill radius. More specifically, ${{ \mathcal R }}_{m}\gg 1$ for r ≲ 0.15 ${R}_{{\rm{H}}}$. Therefore, even if the disk is substantially sub-Keplerian (as is the case, for example, in the simulations of Szulágyi et al. 2016, where v/v_kep ≈ 0.8) the system is guaranteed to reside in a regime where magnetic diffusion plays an essentially negligible role on orbital timescales.

With a rudimentary description of the circumplanetary disk in place, let us now consider the planetary (i.e., central body) parameters. As already mentioned above, we are interested in quantifying the final stages of conglomeration of a Jupiter-mass object. Correspondingly, following Lissauer et al. (2009) and Berardo & Cumming (2017), here we adopt a mass-accretion rate of $\dot{M}={10}^{-3}\,{M}_{\oplus }$ yr⁻¹, a radius of R = 2R_J, and an effective temperature of T_eff = 1500 K, in agreement with the energy boundary condition of the simulated disk in Szulágyi et al. (2016). Cumulatively, these parameters yield a proto-Jovian luminosity of ${L}_{p}\approx 2\times {10}^{-4}\,{L}_{\odot }$, and roughly correspond to the so-called "hot start" initial conditions of giant planet evolution.

A direct consequence of convective energy transport within the planetary interior is the generation of a large-scale magnetic field. Correspondingly, Christensen et al. (2009) have proposed the following equipartition-like relationship that connects the interior heat flux with the magnitude of the generated field:

$$\begin{eqnarray}&&\langle {B}^{2}\rangle /2{\mu }_{0}=c\,{f}_{\mathrm{ohm}}\,\langle \rho {\rangle }^{1/3}{(Fq)}^{2/3}\sim \langle \rho \rangle \,{v}_{\mathrm{conv}}^{2}.\end{eqnarray} \tag{ 5 }$$

In the above expression, c is a constant of proportionality of order unity, f_ohm ≈ 1 is the ratio of Ohmic to total energy dissipation, $\langle \rho \rangle$ is the average density of the field generating region, q = σ_sb T_eff⁴ is the bolometric flux, and F is a quantity related to the ratio of the largest convective length-scale to the temperature scale-height. Furthermore, the surface field is related to the volume-averaged dynamo field through a simple numerical factor $\langle B\rangle /{B}_{{\rm{s}}}\approx 3.5$.

Intriguingly, the relationship (5) successfully connects the Geodynamo, Jupiter's dynamo, as well as fully convective stars, rendering our estimate of the proto-Jovian field an interpolation. For the present-day Jupiter, Christensen et al. (2009) advocate for F = 0.35, which yields c = 1.18. Retaining these numbers, we obtain B_s ≈ 500 Gauss as an estimate for our model planetary surface field strength—a quantity approximately two orders of magnitude in excess of Jupiter's present-day field. Although direct measurements of newly formed giant planets' magnetic fields do not yet exist, it is worth noting that our inferred value is almost an order-of-magnitude smaller than the recently derived magnetic fields of 10–30 M_J brown dwarfs characterized by T_eff ∼ 1000 K (Kao et al. 2016), suggesting that a slightly sub-kiloGauss field is probably not an unreasonable estimate for a highly luminous giant planet (see also Stevenson 1974).

Provided an accretionary flow of magnitude $\dot{M}$, a characteristic radial scale interior to which magnetic effects dominate the dynamics of the system is given by

$$\begin{eqnarray}&&{R}_{t}\sim {\left(\displaystyle \frac{{\pi }^{2}}{2{\mu }_{0}}\displaystyle \frac{{{ \mathcal M }}^{4}}{{ \mathcal G }M{\dot{M}}^{2}}\right)}^{1/7},\end{eqnarray} \tag{ 6 }$$

where ${ \mathcal M }={B}_{{\rm{s}}}\,{R}^{3}$, and the field configuration is assumed to be a pole-aligned dipole.² This well-known expression has been derived by various means in the literature, including balance of the magnetic and ram pressure (Ghosh & Lamb 1979) as well as considerations of angular momentum transport (Ostriker & Shu 1995). However, as pointed out by Mohanty & Shu (2008), apart from a numerical coefficient of order unity, any physical system determined by the quantities inside the parenthesis must have a characteristic length-scale given by Equation (6) by dimensional analysis. Given our fiducial parameters quoted in the previous section, we obtain R_t ∼ 4−5 R_J ≈ 0.006 ${R}_{{\rm{H}}}$.

For radii interior to R_t, the accretionary flow is envisioned to take on a force-free configuration, wherein ${\boldsymbol{v}}\times {\boldsymbol{B}}\to 0$ (Matt & Pudritz 2005; Mohanty & Shu 2008). That is, instead of free-falling, the weakly ionized gas becomes confined to the magnetic field lines, forming an "accretion curtain" around the newly formed planet. While this picture appears qualitatively identical to the one typically invoked for disk-bearing T-Tauri stars (see, e.g., Adams & Gregory 2012), it is crucial to note that the flow direction is reversed.

In the case of young stars, the disk has an overall inward radial velocity, and upon reaching the truncation radius, the gas climbs up the critical field line, eventually arriving to the vicinity of the stellar pole (Batygin & Adams 2013). In the case of the circumplanetary disk, the accretion flow is precipitating upon the planet from a high altitude, implying nearly vertical motion (Tanigawa et al. 2012; Szulágyi et al. 2016). Thus, upon impacting the critical field line, the gas either accretes upon the planet (if it lands interior to the apex of the critical field line) or slides down the magnetic curtain to join the circumplanetary disk. In the latter case, the gas is eventually expelled from the Hill sphere as circumplanetary disk material is recycled back into the nebula (Morbidelli et al. 2014).

The picture described above implies that upon becoming luminous enough to develop a field that is sufficiently strong to partially isolate the vertically precipitating flow (i.e., R_t ≳ 2R_J), a young planet begins to magnetically limit the geometric cross-section through which it can accrete nebular material (Figure 2). Crudely speaking, for a dipole field, this entails a vertical accretion cross-section of ${{ \mathcal A }}_{\mathrm{accr}}\approx \pi {R}_{t}^{2}/{3}^{3/4}$. Therefore, it is plausible that this effect may play an important role in quenching the conglomeration rate of the planet and potentially contribute to a resolution to the terminal mass problem of giant planet formation (Morbidelli et al. 2014). A thorough exploration of this possibility is beyond the scope of the semi-analytic calculations presented in this work, and deserves a more detailed treatment, employing three-dimensional magnetohydrodynamic simulations (Gressel et al. 2013).

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While the region interior to the truncation radius is envisioned to be devoid of gas, the circumplanetary disk that resides exterior to R_t can be thought of as being largely unperturbed by magnetic stresses, on orbital timescales. In other words, the general picture outlined by the hydrodynamical simulations of Szulágyi et al. (2016) remains valid beyond the planetary magnetosphere. Accordingly, let us now calculate the secular angular momentum exchange between the quasi-Keplerian disk and the planet, facilitated by magnetic induction.

In the ${{ \mathcal R }}_{m}\gg 1$ regime appropriate to the problem at hand, field lines that originate within the planet and vertically puncture the circumplanetary disk, B_z, become rapidly³ wound up by Keplerian shear (e.g., Armitage & Clarke 1996). In other words, the orbital motion of the gas induces an azimuthal magnetic field within the disk, B_φ, though flux-freezing. Obviously, growth of B_φ cannot persist indefinitely, and indeed, this process is reset by magnetic reconnection, such that when the pitch angle γ = B_φ/B_z reaches a value in excess of unity, it is impulsively restored back to zero and the cycle repeats (Uzdensky et al. 2002).

In light of the short timescale over which the induction-reconnection cycle unfolds, it is sensible to average over it, and adopt a secular value of γ that is representative of the mean pitch angle of the field lines within the disk (Figure 2). Following Armitage & Clarke (1996), Lai et al. (2011), Spalding & Batygin (2015) and the references therein, we adopt $| \gamma | =1$ everywhere⁴ in the magnetically connected region of the disk (r ≲ 0.15 ${R}_{{\rm{H}}}$). Importantly, the sign of B_φ is determined by the direction of the differential flow, as measured in a frame co-rotating with the planet. Therefore, disk material residing interior to the corotation radius, R_c = (${ \mathcal G }$ M/ω²)^1/3, acts to increase the planetary rotation, while gas exterior to R_c diminishes the planetary spin.

The magnetic torque exerted upon the planet by the disk is given by the sum of the off-diagonal components of the Maxwell stress tensor (Livio & Pringle 1992):

$$\begin{eqnarray}&&{\left(\displaystyle \frac{{dJ}}{{dt}}\right)}_{m}=-\displaystyle \frac{4\pi }{{\mu }_{0}}\int {B}_{\varphi }\,{B}_{z}\,{r}^{2}\,{dr}=-\displaystyle \frac{4\pi }{{\mu }_{0}}\int \gamma \,\displaystyle \frac{{{ \mathcal M }}^{2}}{{r}^{4}}\,{dr},\end{eqnarray} \tag{ 7 }$$

where J is the planetary angular momentum, and the integral is envisioned to run over the magnetically connected domain of the disk. Because the integrand in the above equation falls off rapidly with radius, an outer boundary that is sufficiently large for magnetic coupling at distance to become negligible can be treated as being effectively infinite (quantitatively, replacing 0.15 ${R}_{{\rm{H}}}$ with $\infty$ as the upper bound of the integral in Equation (7) makes virtually no difference for the system under consideration). Thus, adopting this simplification and accounting for the sign change of the induced field across the corotation radius, we obtain the following expressions:

$$\begin{eqnarray}{\left(\displaystyle \frac{{dJ}}{{dt}}\right)}_{m}=\left\{\begin{array}{lc}-\displaystyle \frac{4\,\pi }{3\,{\mu }_{0}}\displaystyle \frac{{{ \mathcal M }}^{2}}{{R}_{t}^{3}} & \omega \gt \sqrt{\displaystyle \frac{{ \mathcal G }\,M}{{R}_{t}^{3}}}\\ \displaystyle \frac{4\,\pi }{3\,{\mu }_{0}}\displaystyle \frac{{{ \mathcal M }}^{2}({R}_{c}^{3}-2\,{R}_{t}^{3})}{{R}_{t}^{3}\,{R}_{c}^{3}} & \omega \leqslant \sqrt{\displaystyle \frac{{ \mathcal G }\,M}{{R}_{t}^{3}}}.\end{array}\right.\end{eqnarray} \tag{ 8 }$$

Equation (8) has a well defined zero-torque solution, which is simply ${R}_{c}={2}^{1/3}{R}_{t}$ (equivalently, $\omega =\sqrt{{ \mathcal G }\,M/2\,{R}_{t}^{3}}$). The characteristic timescale on which the system approaches this equilibrium state from breakup, ω_b, is:

$$\begin{eqnarray}&&{\tau }_{m}=\displaystyle \frac{3\,{\mu }_{0}}{4\,\pi }\displaystyle \frac{I\,M\,{R}_{t}^{3}\,{R}^{2}\,{\omega }_{b}}{{{ \mathcal M }}^{2}}\approx \displaystyle \frac{2}{I}{\left[\displaystyle \frac{{\mu }_{0}\sqrt{{ \mathcal G }{M}^{15}{R}^{7}}}{{\dot{M}}^{6}{{ \mathcal M }}^{2}}\right]}^{1/7},\end{eqnarray} \tag{ 9 }$$

where I is the specific moment of inertia. If we model the interior structure of the young giant planet as a polytropic body with index ξ = 3/2 (appropriate for a fully convective object), then I = 0.21 and for our fiducial parameters, τ_m ≈ 2×10⁴ years—a timescale two orders of magnitude shorter than the typical lifetime of a protoplanetary disk and much smaller than the corresponding value obtained by Takata & Stevenson (1996). We note that the large discrepancy in spin-down timescales obtained here and in previous work stems from the fact that Takata & Stevenson (1996) consider a later epoch in Jovian evolutionary history, when magnetic disk–planet coupling is significantly weaker.

Although Equation (9) clearly indicates that the magnetically facilitated planet–disk angular momentum transfer mechanism is exceedingly efficient, we caution that the mere existence of a magnetized circumplanetary disk is insufficient to resolve the terminal spin problem. This is because taken in isolation, the disk's angular momentum budget is much smaller than that of the planet (note that this situation is reversed in the case of circumstellar disks):

$$\begin{eqnarray}&&\displaystyle \frac{{J}_{\mathrm{disk}}}{J}\lesssim \displaystyle \frac{{\int }_{0}^{\infty }{\int }_{{R}_{t}}^{{R}_{{\rm{H}}}}4\pi \,r\,\rho \,{e}^{\tfrac{-{z}^{2}}{2{h}^{2}}}\,\sqrt{{ \mathcal G }\,M\,r}\,{dr}\,{dz}}{I\,M\,{R}^{2}\,{\omega }_{b}}\sim 0.1.\end{eqnarray} \tag{ 10 }$$

Thus, if the planet and the circumplanetary disk were a closed system, magnetic torques would merely propel the inner regions of the disk to greater orbital radii, while the planetary spin would slow down by a negligible amount. Contrary to this picture, hydrodynamic simulations show that gas is rapidly circulated within the planetary Hill sphere (for example, disk radial velocities of order $\sim {10}^{-3}\mbox{--}{10}^{-2}\,{v}_{\mathrm{kep}}$ are observed in the simulations of Tanigawa et al. 2012; Szulágyi et al. 2016), such that the disk persists in steady-state around the planet, even when magnetic torques are taken into account. As a result, recycling of disk material plays a key role in the successful operation of the magnetic spin-down mechanism, as it connects the circumplanetary disk with its exterior environment and allows the planet to expel its spin angular momentum to the circumstellar nebula.

In addition to magnetic braking, other effects such as continued accretion and gravitational contraction, may contribute to the rotational evolution of a newly formed giant planet (see, e.g., Batygin & Adams 2013). Drawing upon an analogy with the related case of T-Tauri stars, we note that once the planet becomes luminous enough to develop a strong magnetic field, it is likely that direct accretion of angular momentum can be ignored, as the vertically precipitating flow that can be captured within the apex of the critical field line (i.e., accretion curtain) carries a comparatively small amount of angular momentum.⁵ Gravitational contraction, on the other hand cannot be immediately disregarded as being irrelevant.

The radius evolution of a polytropic body that radiates at a constant effective temperature can be approximately modeled by equating the loss of binding energy ${ \mathcal E }=-b\,{ \mathcal G }\,{M}^{2}/R$ to the surface luminosity:

$$\begin{eqnarray}&&\displaystyle \frac{d{ \mathcal E }}{{dt}}=b\displaystyle \frac{{ \mathcal G }\,{M}^{2}}{{R}^{2}}\displaystyle \frac{{dR}}{{dt}}=-4\,{\pi }^{2}{R}^{2}\,{\sigma }_{\mathrm{sb}}\,{T}_{\mathrm{eff}}^{4},\end{eqnarray} \tag{ 11 }$$

where b = 3/(10–2ξ) = 3/7. Adopting R₀ = 2R_J as an initial condition, the solution to the above differential equation is parameterized by the Kelvin–Helmholtz timescale, τ_kh:

$$\begin{eqnarray}R & = & {\left(\displaystyle \frac{b{ \mathcal G }{M}^{2}{R}_{0}^{3}}{b{ \mathcal G }{M}^{2}+12\pi {R}_{0}^{3}{\sigma }_{\mathrm{SB}}{T}_{\mathrm{eff}}^{4}t}\right)}^{1/3}\\ & \to & \max \left[{R}_{{\rm{J}}},{R}_{0}{\left(\displaystyle \frac{{\tau }_{\mathrm{kh}}}{{\tau }_{\mathrm{kh}}+3t}\right)}^{1/3}\right].\end{eqnarray} \tag{ 12 }$$

Importantly, Equation (11) does not account for degeneracy pressure support, which prevents the planet from contracting below a specific radius, close to that of present-day Jupiter (Stevenson & Salpeter 1977). As a consequence, we impose an artificial limitation upon the above expression, which prevents the radius from diminishing beyond its observed value. We note further that in general, gravitational contraction need not end before dissipation of the circumplanetary disk, implying the possibility of mild post-accretion increase in the rotation rate.

With all of the necessary ingredients defined, the differential equation for the planetary spin, ω, takes the following form:

$$\begin{eqnarray}&&2\,I\,M\,R\,\omega \,\displaystyle \frac{{dR}}{{dt}}+I\,M\,{R}^{2}\,\displaystyle \frac{d\omega }{{dt}}={\left(\displaystyle \frac{{dJ}}{{dt}}\right)}_{m},\end{eqnarray} \tag{ 13 }$$

where R, dR/dt and (dJ/dt)_m are given by Equations (12), (11), and (8), respectively. Beyond the assumptions of our rudimentary model, we note that while the characteristic values of $\dot{M}$ and T_eff themselves are somewhat uncertain, their time-evolution is even more obscure, as the accretion process can enhance the interior entropy, leading to a broad range of possible luminosity tracks (Owen & Menou 2016). In our numerical solution of this ODE, we circumvent this ambiguity by holding $\dot{M}$ and T_eff constant at their nominal values, while self-consistently evolving the planetary dipole movement as well as the magnetic truncation radius in accord with Equations (5) and (6). Adopting the breakup velocity as a starting condition for ω, Figure 3 depicts the derived time-series of the planetary rotation period, over a span of 2 Myr.

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Two characteristic phases of evolution are clearly visible in Figure 3. First, on a timescale τ_m (given by Equation (9)), the system approaches the disk-locked equilibrium state, wherein the planetary spin rate is marginally slower than the orbital frequency at the truncation radius. During subsequent evolution that unfolds on the Kelvin–Helmholtz timescale, the planetary spin adiabatically follows the disk-locked state as gravitational contraction forces the dipole moment to diminish, increasing the rotation rate. Upon reaching the current radius of Jupiter, the physical evolution of the system stops and the rotation period becomes frozen-in at 2π/ω ≈ 9 hr. Note that the dramatic changes in the planetary rotation rate over the simulated timespan are driven entirely by the variations in the magnetospheric truncation radius, highlighting our model's sensitive dependence on R_t.

Despite matching the true rotation rate of Jupiter relatively well, we caution that the solution depicted in Figure 3 should be viewed as being essentially illustrative rather than exact. Indeed, there exists a number of intricacies and uncertainties that Equation (13) simply does not account for. In particular, recall that we have ignored the inevitable decrease of $\dot{M}$ that must come about in a waning circumstellar nebula, the timing of giant planet conglomeration relative to disk dissipation, as well as the decline of T_eff that ensues as the planet contracts. Nevertheless, the presented calculation demonstrates the efficiency of the magnetic braking mechanism within the context of planet–disk interactions, and offers qualitative insight into the physical processes that operate concurrently with giant planet formation and regulate angular momentum exchange within the planet's local sphere of influence.