Low and highorder gravitational harmonics of rigidly rotating Jupiter
Universität Rostock, Institut für Physik, 18051 Rostock, Germany
email: nadine.nettelmann@unirostock.de
Received: 11 July 2017
Accepted: 19 August 2017
The Juno Orbiter has provided improved estimates of the even gravitational harmonics J_{2} to J_{8} of Jupiter. To compute higherorder moments, new methods such as the concentric Maclaurin spheroids (CMS) method have been developed, which surpass the commonly used theory of figures (ToF) method in accuracy. This progress raises the question whether ToF can still provide a useful service for deriving the internal structure of giant planets in the solar system. In this paper, I apply both the ToF and the CMS method to compare results for polytropic Jupiter and for the physical equation of state H/HeREOS.3based models. An accuracy in the computed values of J_{2} and J_{4} of 0.1% is found to be sufficient in order to obtain the core mass safely within 0.5 M_{⊕} numerical accuracy and the atmospheric metallicity within about 0.0004. ToF to the fourth order provides that accuracy, while ToF to the third order does not for J_{4}. Furthermore, I find that the assumption of rigid rotation yields J_{6} and J_{8} values in agreement with the current Juno estimates, and that higherorder terms (J_{10} to J_{18}) deviate by about 10% from predictions by polytropic models. This work suggests that ToF 4 can still be applied to infer the deep internal structure of giant planets, and that the zonal winds on Jupiter reach less deep than 0.9 R_{J}.
Key words: planets and satellites: individual: Jupiter / planets and satellites: interiors
© ESO, 2017
1. Introduction
The theory of figures (ToF; Zharkov & Trubitsyn 1978) to third or fourth order, hereafter labeled respectively ToF 3 and ToF 4, is commonly used to compute the gravity field of the gas giant planets in the solar system (e.g., Saumon & Guillot 2004; Helled 2011; Leconte & Chabrier 2012; Nettelmann et al. 2012; Helled & Guillot 2013; Miguel et al. 2016). While a theory to nth order allows for computation of the gravity field in terms of the gravitational harmonics up to J_{2n} only, Jupiter’s gravity field before the current Juno mission was also measured up to J_{6} only. Thus, third or fourth order theories seemed sufficient. Moreover, it is the loworder harmonics J_{2} and J_{4} that are particularly sensitive to the internal density distribution; they allow for derivation of the interior structure parameters core mass and envelope metallicity. Hence for a long time, the observational data of gravity field theories (e.g., ToF) and planetinterior parameters of interest formed a closed system.
This convenient situation has changed with the arrival of the Juno spacecraft at Jupiter. Juno’s sensitivity limit makes it possible to measure the rigidrotation contribution to the gravitational harmonics up to J_{14} (Kaspi et al. 2010). Highorder moments yield clues to the properties of the zonal winds as the flows influence the density distribution, which in turn is the source function of the gravitational potential. Differential rotation due to zonal flows is predicted to entirely dominate the J_{2n} for n ≥ 14, while it is predicted to be within a factor of ten of the prediction for a rigidly rotating planet for J_{8}−J_{12} (Hubbard 1999; Kaspi et al. 2010; Cao & Stevenson 2017). Since the wind contribution is obtained by subtracting the theoretical values for a rigidly rotating planet from the observed ones (), it is also important to have accurate knowledge of the rigidrotation contribution.
For that purpose, Hubbard (2013, hereafter H13), developed the concentric Maclaurin spheroids (CMS) method. This method yields demonstratively good agreement with the exact Bessel solution for an n = 1 polytrope model of Jupiter. Deviations have been found to be about 5 × 10^{5} in J_{2} to 2 × 10^{4} in J_{20} (Wisdom & Hubbard 2016), or to be of the order of 2 × 10^{3} (Cao & Stevenson 2017). However, comparison of the exact Bessel solution to the ToF 3 results (H13; Wisdom 1996) has led to the conclusion that ToF 3 is insufficiently accurate for modeling Jupiter (Wisdom 1996). This raises the question of what accuracy in the loworder moments is desired for inferring Jupiter’s internal density distribution, and which methods can provide that.
Fig. 1 Resulting J_{2} values of polytropic models of rigidly rotating Jupiter for q = 0.089195487 (left) as in H13 and Wisdom & Hubbard (2016), and for q = 0.088570679 (right) as in Wisdom (1996). Blue symbols: using ToF to fourth order, blue lines: respective fit curves, light blue: same as blue but using ToF 3, red symbols: using CMS method. Reference values are in black; black circle: CMS results of H13 for N = 512, horizontal black lines: unknown value of N, in particular: thick black lines: Bessel/CLC results of Wisdom & Hubbard (2016), thin black lines: ToF 3 results of H13 and Wisdom (1996). The vertical black arrow shows the current Juno uncertainty of J_{2} (Folkner et al. 2017), here arbitrarily placed. The xaxis is number of radial grid points N. 

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Fig. 2 Same as Fig. 1 but for J_{4}. The ToF 3 result of Hubbard (2013) exceeds the shown range of J_{4} values. 

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The classical view of a Jupiterlike gas giant is that of a welldefined core embedded into an H/Herich envelope, in which case one can search for the mass of the core and the heavy element mass fraction of the envelope and use the measured planet radius and J_{2} value to determine both (Militzer et al. 2008). However, recent gas giant formation models challenge that simple picture. Simultaneous accretion of gas and planetesimals might naturally lead to a gasenriched, that is, diluted core (Venturini et al. 2016), where the metallicity is predicted to decrease outward as a result of the slow convective timescale compared to the accretion rate during formation (Helled & Stevenson 2017); a diluted core may remain permanent due to the inhibition of convection once a compositional gradient has become established (Vazan et al. 2016). Diluted cores have been found to enhance the predicted atmospheric metallicity of Jupiter models (Fortney & Nettelmann 2010).
In this paper, the uncertainty in the computed values for the loworder harmonics J_{2}, J_{4}, J_{6} due to application of ToF 3 and ToF 4 is estimated by using the n = 1 polytrope model (Sect. 2); the corresponding uncertainty in the derived core mass and envelope metallicity of Jupiter is estimated in Sect. 3. Finding this uncertainty to be small for ToF 4, I use this method in Sect. 4 to compute physical equations of state (EOS)based Jupiter models that are designed to match the loworder harmonics J_{2} and J_{4} from Juno’s first two lowperiapse polar orbits around Jupiter (Folkner et al. 2017). Models are presented both for solid cores and for diluted cores, as well as for deep zonal wind corrections as proposed by Cao & Stevenson (2017). For some of the Jupiter models, I compute the highorder moments using the CMS method (Sect. 5), thereby providing the first prediction of the highorder J_{2n} values for a model of an adiabatic, rigidly rotating Jupiter that matches the measured loworder moments. Conclusions are in Sect. 6. In Appendix A my implementation of the CMS method is validated for the linear density case, while in Appendix B the ToF coefficients are provided up to the fourth order.
2. Polytropic models
In this section, n = 1 polytropic models are computed for GM_{J} = 12.6686536 × 10^{16}m^{3}/ s^{2}, equatorial radius R_{eq} = R_{J} = 71 492 km, and for two different rotation rates as represented by q = 0.0891954870 (Wisdom & Hubbard 2016) and q = 0.08857067907 (Wisdom 1996), where . I apply ToF to the third and fourth order as well as the CMS method. For both methods an iterative procedure is required to ensure the total mass is conserved, and that for the thus specified value of K in the polytropic relation P = Kρ^{2} hydrostatic balance holds.
With ToF I calculate the density at grid point i using , while with CMS method (H13), except for i = 0 where ρ_{0} = P_{0} = 0 in their respective units. As I find the dependence on the number of radial grid points, N, to be strong, I plot the resulting values of J_{2} (Fig. 1), J_{4} (Fig. 2), and J_{6} (Fig. 3) against N.
Fig. 3 Same as Figs. 1 and 2 but for J_{6} and results for different qvalues merged into a single panel, distinguished by respectively solid lines/circles and dashed lines/diamonds. The ToF 3 result of Hubbard (2013) exceeds the shown range of J_{6} values. 

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Compared to the exact Bessel solution (Wisdom & Hubbard 2016), the CMS method performs best and ToF 3 worst. In particular, ToF 3 underestimates  J_{4}/ 10^{6}  by one to two times (depending on the implementation) the preJuno 1σ error bar of ~2 (Miguel et al. 2016), and therefore predicts a higher atmospheric metallicity for Jupiter than ToF 4 does (Nettelmann et al. 2012).
ToF 4, on the other hand, performs much better: the difference in J_{4} to the exact Bessel solution amounts to only about 15% of the total preJuno error bar, and the differences in both J_{4} and J_{6} are still smaller than the uncertainties of the current Juno data. The influence of the error in J_{4} due to the application of ToF 4 on the predicted envelope metallicity and core mass of Jupiter can be considered negligible. In the following section, I investigate whether this is also the case for J_{2}, the error bar of which is 5× the current Juno estimate of 2 × 0.272 × 10^{6} (Fig. 1).
3. Interior models and J_{2}
The observed value of J_{2} allows for insight into the internal structure of Jupiter, as different internal density distributions may yield different values of J_{2} to be compared against the observed one. In this section I investigate how sensitive that dependence is. In particular, I consider the resulting uncertainty in the derived core mass (M_{core}) and atmospheric metallicity (Z_{atm}) due to the technical uncertainty in J_{2}, which results from applying ToF 4 to compute the gravitational harmonics. For this purpose, simple models are computed for which I assume a constant metallicity throughout Jupiter’s envelope. Although further details of the procedure do not influence the resulting quantities we are interested in (the uncertainties), I give them for completeness: the envelope is separated into an outer, Hepoor part of helium abundance Y_{atm} = 0.238 in agreement with the Galileo entry probe value, and a Herich inner envelope that accounts for the remaining helium to yield a total He/H mass ratio of 0.275 in agreement with estimates for the protosolar cloud. The transition takes place at pressure P_{trans} = 8 Mbar. The envelope adiabat runs through the temperaturepressure point of 423 K at 22 bars as measured by the Galileo entry probe. At the outer boundary at one bar this yields T_{1} = 170 K, which I adopt as the outer boundary condition for the Jupiter models. Figure 4 shows the resulting uncertainties in M_{core} and Z_{atm} as a function of the assumed value of J_{2}.
Fig. 4 Uncertainty in the derived values for core mass and atmospheric metallicity due to the assumed uncertainty in J_{2} for interior models assuming constant envelopeZ. For ΔJ_{2} about 10 × the current Juno uncertainty, corresponding to 2 × the estimated error from applying ToF 4 (bottom panel) the core mass uncertainty amounts to ~ 0.04 M_{⊕} (middle panel), while the uncertainty in Z_{atm} is found to be less than 0.0002 (upper panel). These uncertainties are small. 

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Fig. 5 Same as Fig. 4 but for J_{4} (lower panel) and J_{6} (upper panel). 

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According to Fig. 4, the error in J_{2} of about 2 × 10^{4} due to applying ToF 4 maps onto an uncertainty of 0.02 M_{⊕} in Jupiter’s core mass and 0.0001 in Z_{atm}. Furthermore, an uncertainty of 0.1% in J_{2} (twice the horizontal length of gray lines) would imply an uncertainty of ~ 0.1 M_{⊕} in core mass and 0.0004 in Z_{atm}. Thus, the uncertainties due to applying ToF 4 can be considered tiny compared to the accuracy in internal structure properties we are interested in, which is about 10% (e.g., ~ 1 M_{⊕} in core mass). Moreover, the uncertainty from this source of error is clearly smaller than the uncertainty due to the material input physics like the EOS, which is at best of the order of 1%. Furthermore, the additional error in J_{4} and J_{6} which results from the uncertainty in J_{2} due to applying ToF 4 amounts to only 0.04% for J_{4} and 0.03% for J_{6} (Fig. 5). From Figs. 4 and 5 I therefore conclude that ToF 4 yields quantitatively useful density distributions for Jupiter.
4. Results for Jupiter
In this section I construct models that aim to match the tight current Juno constraints on J_{2} and J_{4} and are based on H/HeREOS.3 (Becker et al. 2014, hereafter B14). In Sect. 4.1 I assume rocky cores and rigid rotation, while in Sect. 4.2 I assume diluted cores or take into account the shift due to winds.
4.1. Models with solid cores and rigid rotation
The models in this section are threelayer models and constructed as in Nettelmann et al. (2012), hereafter N12. The only but important difference to the models of Sect. 3 is that threelayer models allow for different heavy element abundances in the two envelopes, so that two free parameters (Z_{1} = Z_{atm} in the outer and Z_{2} in the inner envelope) are available for adjusting the two loworder harmonics J_{2} and J_{4}. If this can be achieved and if in addition Z_{1} ≥ 2 × solar, consistent with the observed heavy noble gas abundances in Jupiter’s atmosphere, I consider a model as acceptable for Jupiter.
Fig. 6 ToF 4 based threelayer Jupiter models that are designed to match observed J_{2} and J_{4} values. This work: black solid, B14: yellow, N12: green. Measured atmospheric particle abundances of the elements as labeled are scaled by their protosolar particle abundance values (Lodders 2003). I use Z_{solar} = 0.015. 

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Figure 6 shows these new models as a function of P_{trans}. They are similar to the ToF 4 based models of N12, who applied HREOS.2 and HeREOS.1, and of B14 who applied HREOS.3 and HeREOS.3 as in this work. The biggest differences are the narrower range in possible transition pressure and the lower Z_{1} values compared to B14. The latter is mainly a direct consequence of reducing  J_{4}/ 10^{6}  from 589 to 586.6. The lower Z_{1} values also tend to reduce J_{2}, requiring more heavy elements in the deep interior to compensate for that. Slightly higher Z_{2} values then leave less mass to build the core, so that finally a smaller set of models (a smaller range of P_{trans} values for which M_{core} ≥ 0) is found. In contrast, the difference between these models and the N12 results was mainly due to differences in the helium EOS at outer envelope pressures. These new results confirm that ab initio H/HeEOSs yield rather low atmospheric metallicities for Jupiter. Compared to ≲ 1 × solar (Hubbard & Militzer 2016; Wahl et al. 2017b), ≲ 2.5 × solar (N12), ≲ 3 × solar (B14), I here obtain Z_{atm} ≲ 2 × solar, out of which acceptable models have P_{trans} = 6–7 Mbar.
An inaccuracy in J_{4} of about 2.4/600 (0.4%) (compare black and yellow curves in Fig. 6) seems to induce a rather large uncertainty of ΔM_{core} = 2 M_{⊕} in core mass; but a 0.1% uncertainty in J_{4} might still lead to ΔM_{core} = 0.5 M_{⊕} for threelayer models. However, this estimate is probably a far upper bound as the models in B14 were computed with a smaller number of grid points of N ~ 2000 compared to ~12 000 in this work.
Fig. 7 (Left panel) Resulting ToF 4based J_{6} values of Jupiter models (magenta). The J_{4} value of these models (magenta, right panel) was adjusted to match the Juno J_{4} measurement. Black diamonds: Juno measurements, blue error bars: estimated uncertainty due to applying ToF 4 according to Figs. 1–3. 

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Figure 7 compares the resulting J_{6} value of the ToF 4based Jupiter models that match the Juno J_{2} and J_{4} values to the Juno measurement of J_{6}. Models with P_{trans} = 4–7 Mbar are within the observational uncertainty of J_{6}, while models with lower transition pressures are within 2σ of the observational uncertainty. The computational error from ToF 4 is much smaller than that.
4.2. Models with diluted cores or zonal winds
Ab initio H/He EOSbased Jupiter models with rockice cores and without zonal winds become notoriously low in atmospheric heavy element abundances. On the other hand, diluted cores have been found to enhance Z_{atm} by up to 50% (Fortney & Nettelmann 2010), while zonal winds directly affect the J_{2} and J_{4} values to be matched by rigidly rotating models (Militzer et al. 2008; Cao & Stevenson 2017) and thus also influence the derived Z_{atm} value. While precise predictions on the dynamic contributions ΔJ_{2n} to the observed values depend on the differential rotation pattern and their mathematical description (Kaspi et al. 2010; Zhang et al. 2015; Cao & Stevenson 2017), it is predicted that the effect on the loworder J_{2n} increases with the depth of the winds (Kaspi et al. 2010; Cao & Stevenson 2017) and that the effect on the loworder J_{2n} is small and in the direction of reducing the absolute J_{2n} values (Hubbard 1999; Kaspi et al. 2010; Cao & Stevenson 2017). Here I calculate Jupiter models as in Sect. 4.1 but by assuming a diluted core of rock mass fraction Z_{3, Rocks} = 0.2, the rest being inner mantle material, and by including zonal wind corrections as proposed by Cao & Stevenson (2017) for halfamplitude width (HAWD) values of 0.8 and 0.9. The latter quantity is defined as the distance to the rotation axis where the azimuthal wind velocity has weakened by a factor of two from its maximum value farther out.
Fig. 8 Jupiter models with diluted cores of central rock mass fraction Z_{3, Rocks} = 0.2 (black dashed) or with zonal winds according to HAWD = 0.9 (solid blue). The solid black curves are the same as in Fig. 6. I use Z_{solar} = 0.015. 

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As shown in Fig. 8, the zonal wind corrections lead to lower values in Z_{1} and M_{core}. This is not a surprise, since the absolute values of J_{2} and J_{4} are reduced and thus demand a smaller mass density in the part of the planet planet where they are most sensitive, which is near P ~ 1 Mbar in the outer envelope. This behavior is in line with the observation of Militzer et al. (2008) who, in order to enhance the resulting envelope Z value, suggest zonal wind effects on J_{2} and J_{4} in the opposite direction of what zonal wind models that fit the observed wind speeds predict.
For HAWD = 0.8 I do not obtain any acceptable Jupiter model because the ΔJ_{2n} are too large: both Z_{1} and M_{core} would become negative. For HAWD = 0.9, there is a restricted range of solutions at P_{trans} = 3–5.5 Mbar, for which Z_{1} barely reaches 1.5 × solar. Adiabatic H/HeREOS.3based Jupiter models thus suggest the vertical extent of the winds to be less than 0.9 R_{J} (~7000 km).
Assuming a diluted core and adiabatic envelopes, the Z_{1} value can be lifted, but only to less than its maximum value obtained for coreless models. The enhancement in Z_{1} can indeed reach up to 50% for the largest core mass found here, but then the base Z_{1} value is small anyway. Therefore, as Fig. 8 shows, diluted cores do not significantly enhance Z_{atm} for H/HeREOSbased models, but are helpful for larger core models (Wahl et al. 2017b) such as those obtained with the EOS of Militzer & Hubbard (2013).
5. Highorder gravitational harmonics
To compute the highorder gravitational harmonics of models that match the observed Juno values for J_{2} and J_{4}, I use the density distributions of the models from Sect. 4 and apply the CMS method to them^{1}. For that purpose, I convert the ToFbased density profile as a function of the mean radius of an equipotential surface to a density profile as a function of the equatorial radius of the same equipotential surface using the ToF 4based figure functions. Then I reduce the number of radial grid points from N ~ 12 000 to N ~ 1000 by assuming a radial spacing that decreases continuously from the middle to the boundaries. At layer boundaries, the jump in density is conserved as illustrated in Fig. 9.
Fig. 9 Conversion of the ToF 4based density profile (black) to a discrete grid as required by the CMS method, here illustrated for N = 50 (blue) and N = 100 (green). 

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Highorder moments of different models for rigidly rotating Jupiter.
Finally, Table 1 presents my results for the low and highorder J_{2n} values of two models from Sect. 4.1, that is, for P_{trans} = 3 Mbar (model J173a) and 6 Mbar (model J176a), and for one model that accounts for deep zonal winds through the corrections to J_{2} and J_{4} for HAWD = 0.9 from Sect. 4.2 (model J174z). Resulting moments of order ≥6 are not affected by the above described procedure within the number of digits given in Table 1. This is shown by model variant (b), where the J_{2} value to be fitted was shifted by the difference ToF 4 minus CMS according to model variant (a). The results are compared to the exact polytrope solution (Wisdom & Hubbard 2016), to the DFTMD7.13 Jupiter model of Hubbard & Militzer (2016), to the Juno measurements of Folkner et al. (2017), and to the same data but corrected for deep zonal winds as proposed by Cao & Stevenson (2017).
Perhaps most interestingly, the resulting values for J_{6} and J_{8} for a rigidly rotating Jupiter are within the current Juno observational error bars. This may indicate that the winds are shallow. The J_{8} value of model J174z is also within the observational error bar and the reduction of its  J_{8}  value by a few percent is much less than the few 10% estimate of Cao & Stevenson (2017), whose ΔJ_{8} estimate peaks for HAWD = 0.9 compared to deeper (0.8) or shallower (0.975) depths. Furthermore, the highorder J_{2n} values of n = 1 polytropic Jupiter differ by about 10% from the physical EOSbased Jupiter models. Thus it is important to provide the latter class of models as done in this work and in Hubbard & Militzer (2016).
6. Conclusions
To infer Jupiter’s internal density distribution, the relative accuracy in the computed values of J_{2} and J_{4} should be of the order of 0.1% (Figs. 4 and 6). According to n = 1 polytropic models, ToF 4 can provide this accuracy, while ToF 3 can do so only for J_{2} (Figs. 1 and 2). The error in J_{2} (J_{4}, J_{6}) due to applying ToF 4, as measured by the difference to the exact solution, is about 5 × (1/2 ×, 1/3 ×) the current Juno uncertainties for these parameters. I conclude that these uncertainties are nevertheless sufficiently small for predicting Jupiter’s internal density distribution. Other uncertainties, such as the thermal state, perhaps as a result of helium rain, may induce larger unknowns in our understanding of Jupiter (Nettelmann et al. 2015; Hubbard & Militzer 2016; Mankovich et al. 2016; Wahl et al. 2017b).
The computed values of J_{6} and J_{8} of rigidly rotating Jupiter suggest that zonal winds are restricted to regions well above a depth of 7000 km (0.9R_{J}). Furthermore, application of the more accurate CMS method in combination with the physical EOS H/HeREOS.3 is found to yield higherorder  J_{2n}  values that are 10% higher than the prediction from the polytropic model (Table 1).
Still, the internal structure of Jupiter remains poorly constrained. Further insight might be gained from a Juno measurement of the fluid Love number k_{2} and its consideration in threedimensional models for the gravity field (Wahl et al. 2017a).
One could of course use the CMS method right from the start; however, my current implementation of the CMS method runs orders of magnitudes slower than my implementation of the ToF method. Given the number of iterations necessary to fit both J_{2} and J_{4} according to the procedure outlined in Nettelmann (2011), I evaluated the accuracy gained as being not worth the enormous computational extra effort.
In the representation by equipotential surfaces, r<l′ can happen for the external field and r>l′ for the internal field. This does not pose a problem here (Zharkov & Trubitsyn 1978) as long as the mvalue is sufficiently small (Hubbard et al. 2014).
Acknowledgments
I thank Naor Movshovitz for sharing results for linear density models with the CMS method, Ronald Redmer and Ravit Helled for inspiring discussions, and the Referee and the Scientific Editor for helpful comments that improved the paper. This work was supported by the DFG grant NE1734/11 of the German Science Foundation.
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Appendix A: Linear density models with CMS
Hubbard (2013) provides the gravitational harmonics for a linear density model with Jupiterlike parameters and N = 128. I use that model to test my implementation of the CMS method. The agreement is excellent for the given number of digits (seven in H13; see Table A.1). However, a model with only N = 128 is not yet converged as the same linear density model with N = 512 shows. Therefore, in order to obtain this good agreement with the linear density model of H13, I had to choose the spacings in equatorial radius dλ_{i} and the dimensionless density jumps δ_{i} exactly as in H13. In particular (N. Movshovitz, pers. comm.), λ_{0} = 1, λ_{N} = 0, δ_{0} = δ_{N} = 0, dλ = 1/(N−1), λ_{1} = λ_{0} − dλ/ 2, λ_{i} = λ_{i−1}−dλ for i> 1, and δ_{i} = dλ_{i} for i ≥ 1. All other parameters were chosen as in H13, in particular q = 0.088822426, R_{eq} = 71 492 km, and GM = 126 686 536km^{3}/ s^{2}.
CMS results for linear density model.
Appendix B: ToF to fourthorder coefficients
I summarize the theory of figures of Zharkov & Trubitsyn (1978) and then give the coefficients up to the fourth order. Consider a spheroidal planet in hydrostatic equilibrium of density distribution ρ(r,ϑ) which is symmetric with respect to the axis of rotation and the equatorial plane. As a result, there is no dependence on azimuthal angle ϕ, and only even indices in the spherical harmonics expansions survive. In this twodimensional problem, a surface of constant total potential U only depends on polar angle ϑ. Different such surfaces r_{l}(ϑ) are labeled by the level parameter l. In the ToF method according to Zharkov & Trubitsyn (1978), l is taken to be the mean radius of the respective equipotential surface as defined by the condition of equal volume, . Furthermore, any dependence on (r,ϑ) is replaced by dependence on (l,ϑ) through the expansion of r_{l}(ϑ) into a series of Legendre polynomials P_{n}(cosϑ) according to (B.1)where the s_{2n}(l) are the figure functions. The firstorder deviation from a spherical shape is described by s_{2}, while s_{0} can be determined with the help of the equalvolume condition to (B.2)The figure functions s_{2n} are of nth order except s_{0} which is of second order. In the following, I abbreviate the expression in parenthesis in Eq. (B.1) by (1 + Σ) and set μ = cosϑ.
The total potential is composed of the gravitational potential while the centrifugal potential reads . In the ToF it is convenient to capture the centrifugal term due to the planetary rotation of angular rotation rate ω by the small parameter , where R_{m} is the mean radius of the outermost level surface. After expanding V and Q into series of Legendre polynomials and replacing r by Eq. (B.1), one can write (B.3)where denotes the mean density . On equipotential surfaces, dU/ dθ = 0 and thus A_{2k} ≡ 0 for k> 0. This property is used to determine the s_{2n}, while A_{0} yields the total potential. One finds (B.4)}The functions S_{n} and will be defined below.
Appendix B.1: From V to D_{n}, D and further to S_{n}, S, and J_{n}
The gravitational potential at a location (r,ϑ) in the planet separates into an external potential r>r′ due to the mass distribution ρ(r′,ϑ′) interior to r and an internal potential r<r′ due to the mass distribution ρ(r′,ϑ′) exterior to r. The multipole expansion of V reads (B.5)Using Eq. (B.1), the volume integrals D_{n} of the external and of the internal gravity field expansion take the form^{2}(B.6)With z: = l/R_{m}, their dimensionless form is defined as (B.7)and can be written as (B.8)After application of partial integration and assuming dρ/ dz to be finite, the S_{n}, adopt the convenient form for numerical evaluation (B.9)with (B.10)By expressing powers of (1 + Σ) in terms of the binomial series expansions, and by further expanding powers of Σ into linear series of Legendre polynomials and by making use of for n ≠ m, the integrals in Eqs. (B.10) can be solved analytically. The results for f_{n}(z) and are provided in Eqs. (B.16) and (B.17). Accordingly, the integrals S_{2n} and are of nth order. Finally, the gravitational harmonics are obtained as (B.11)
Appendix B.2: Coefficients in A_{2n} for computing the s_{2n}
Below, I give the coefficients that are of fourth order or lower after multiplication with m, S_{2n}, or as occurring in the respective equations. They were generated by a C++ program written by myself in 2004, (B.12)To lowest order, s_{2} ≈ −m/ 3, thus s_{2} is of first order in m, (B.13)To lowest order, s_{4} ~ m × s_{2}, thus s_{4} is of second order in m,
(B.14)To lowest order, s_{6} ~ m × s_{4}, thus s_{6} is of third order in m. (B.15)To lowest order, s_{2n} ~ m^{n}.
Appendix B.3: Coefficients for computing the f_{2n} and f′_{2n}
(B.17)}Numerical values for the s_{2n} and J_{2n} for a given barotrope ρ(P) can be obtained through an iterative procedure. For given values of the J_{2n} and s_{2n}, which can initially be zero, the density distribution ρ(l) is computed by numerical integration of the hydrostatic balance equation dP/ dl = −ρ(P)dU/ dl using Eq. (B.4), and by integration of the mass conservation equation dm/ dl = 4πl^{2}ρ(l). Given then ρ(l), new figure functions are repeatedly calculated until convergence using Eqs. (B.12) to (B.15), and then the J_{2n} are calculated using Eq. (B.11). Converged J_{2n} values for a given barotrope require about six iterations of this procedure.
All Tables
All Figures
Fig. 1 Resulting J_{2} values of polytropic models of rigidly rotating Jupiter for q = 0.089195487 (left) as in H13 and Wisdom & Hubbard (2016), and for q = 0.088570679 (right) as in Wisdom (1996). Blue symbols: using ToF to fourth order, blue lines: respective fit curves, light blue: same as blue but using ToF 3, red symbols: using CMS method. Reference values are in black; black circle: CMS results of H13 for N = 512, horizontal black lines: unknown value of N, in particular: thick black lines: Bessel/CLC results of Wisdom & Hubbard (2016), thin black lines: ToF 3 results of H13 and Wisdom (1996). The vertical black arrow shows the current Juno uncertainty of J_{2} (Folkner et al. 2017), here arbitrarily placed. The xaxis is number of radial grid points N. 

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In the text 
Fig. 2 Same as Fig. 1 but for J_{4}. The ToF 3 result of Hubbard (2013) exceeds the shown range of J_{4} values. 

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In the text 
Fig. 3 Same as Figs. 1 and 2 but for J_{6} and results for different qvalues merged into a single panel, distinguished by respectively solid lines/circles and dashed lines/diamonds. The ToF 3 result of Hubbard (2013) exceeds the shown range of J_{6} values. 

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In the text 
Fig. 4 Uncertainty in the derived values for core mass and atmospheric metallicity due to the assumed uncertainty in J_{2} for interior models assuming constant envelopeZ. For ΔJ_{2} about 10 × the current Juno uncertainty, corresponding to 2 × the estimated error from applying ToF 4 (bottom panel) the core mass uncertainty amounts to ~ 0.04 M_{⊕} (middle panel), while the uncertainty in Z_{atm} is found to be less than 0.0002 (upper panel). These uncertainties are small. 

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In the text 
Fig. 5 Same as Fig. 4 but for J_{4} (lower panel) and J_{6} (upper panel). 

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In the text 
Fig. 6 ToF 4 based threelayer Jupiter models that are designed to match observed J_{2} and J_{4} values. This work: black solid, B14: yellow, N12: green. Measured atmospheric particle abundances of the elements as labeled are scaled by their protosolar particle abundance values (Lodders 2003). I use Z_{solar} = 0.015. 

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In the text 
Fig. 7 (Left panel) Resulting ToF 4based J_{6} values of Jupiter models (magenta). The J_{4} value of these models (magenta, right panel) was adjusted to match the Juno J_{4} measurement. Black diamonds: Juno measurements, blue error bars: estimated uncertainty due to applying ToF 4 according to Figs. 1–3. 

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In the text 
Fig. 8 Jupiter models with diluted cores of central rock mass fraction Z_{3, Rocks} = 0.2 (black dashed) or with zonal winds according to HAWD = 0.9 (solid blue). The solid black curves are the same as in Fig. 6. I use Z_{solar} = 0.015. 

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In the text 
Fig. 9 Conversion of the ToF 4based density profile (black) to a discrete grid as required by the CMS method, here illustrated for N = 50 (blue) and N = 100 (green). 

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In the text 