EQUATORIAL ZONAL JETS AND JUPITER's GRAVITY

2015

ApJ

TITLEThermal-gravitational wind equation for the wind-induced gravitational signature of giant gaseous planets: mathematical derivation, numerical method, and illustrative solutions AUTHORS Schubert, G; Zhang, Keke; Kong, Dali; et al. ABSTRACTThe standard thermal wind equation (TWE) relating the vertical shear of a flow to the horizontal density gradient in an atmosphere has been used to calculate the external gravitational signature produced by zonal winds in the interiors of giant gaseous planets. We show, however, that in this application the TWE needs to be generalized to account for an associated gravitational perturbation. We refer to the generalized equation as the thermalgravitational wind equation (TGWE). The generalized equation represents a two-dimensional kernel integral equation with the Green's function in its integrand and is hence much more difficult to solve than the standard TWE. We develop an extended spectral method for solving the TGWE in spherical geometry. We then apply the method to a generic gaseous Jupiter-like object with idealized zonal winds. We demonstrate that solutions of the TGWE are substantially different from those of the standard TWE. We conclude that the TGWE must be used to estimate the gravitational signature of zonal winds in giant gaseous planets. Key words: planetary systemsplanets and satellites: gaseous planetsplanets and satellites: interiors q without accounting for the associated gravitational perturbation. The observed zonal gravitational coefficients J n in Equation (1) would compri-sethe two parts J J J , n n n static dyn = +Dwhere J n static is caused by the effect of rotational distortion while J n dyn

Section: Introductionmentioning

confidence: 99%

Thermal-Gravitational Wind Equation for the Wind-Induced Gravitational Signature of Giant Gaseous Planets: Mathematical Derivation, Numerical Method, and Illustrative Solutions

2015

ApJ

“…On the basis of the assumptions that the rotational effect upon the Jovian shape is negligibly small and that the observed cloud-level zonal winds extend on the cylinders, but decay exponentially in the radial direction, Kaspi et al (2010) solved the thermal-wind equation to estimate the wind-induced density anomaly and the corresponding gravitational correction DJ n with n even and  n 2. By assuming that Jupiter is in the shape of an oblate spheroid and that the observed cloud-level zonal winds extend all the way on the cylinders from the northern to the southern hemisphere without being blocked by the Jovian magnetic field, Kong et al (2013a) computed the variation of the Jovian gravitational coefficients ΔJ 2 , ΔJ 4 ,..., ΔJ 12 caused by the effect of the zonal winds, while Kong et al (2014) revealed the dominance of the equatorial zonal jets in determining the high-order gravitational coefficients. Furthermore, the Jovian magnetic field is generated by convection-driven motion in its deep metallic region that must be non-axisymmetric, but whose amplitude and structure are unknown; Kong et al (2016) discussed a promising way of probing the Jovian convective dynamo via its effect on the external non-axisymmetric gravitational field.…”

Section: Introductionmentioning

confidence: 99%

A Fully Self-Consistent Multi-Layered Model of Jupiter

2016

ApJ

We construct a three-dimensional, fully self-consistent, multi-layered, non-spheroidal model of Jupiter consisting of an inner core, a metallic electrically conducting dynamo region, and an outer molecular electrically insulating envelope. We assume that the Jovian zonal winds are on cylinders parallel to the rotation axis but, due to the effect of magnetic braking, are confined within the outer molecular envelope. We also assume that the location of the molecular-metallic interface is characterized by its equatorial radius HR e , where R e is the equatorial radius of Jupiter at the 1 bar pressure level and H is treated as a parameter of the model. We solve the relevant mathematical problem via a perturbation approach. The leading-order problem determines the density, size, and shape of the inner core, the irregular shape of the 1 bar pressure level, and the internal structure of Jupiter that accounts for the full effect of rotational distortion, but without the influence of the zonal winds; the next-order problem determines the variation of the gravitational field solely caused by the effect of the zonal winds on the rotationally distorted non-spheroidal Jupiter. The leading-order solution produces the known mass, the known equatorial and polar radii, and the known zonal gravitational coefficient J 2 of Jupiter within their error bars; it also yields the coefficients J 4 and J 6 within about 5% accuracy, the core equatorial radius R 0.09 e and the core density r =´-2.0 10 kg m corresponding to 3.73 Earth masses; the next-order solution yields the wind-induced variation of the zonal gravitational coefficients of Jupiter.

“…An important objective of the Juno spacecraft is to probe the extent of penetration of the zonal winds into the interior of Jupiter by accurately measuring their effects on its gravitational field with unprecedentedly high precision. Since the high-order coefficients J n with even n are found to be nearly independent of the depth of the zonal winds in the non-equatorial regions (Kong et al 2014), and since the size of J n with odd n are directly related to the depth of the equatorially antisymmetric zonal winds (Kaspi 2013), accurately determining the odd coefficients J n in equation (1) would play a critical role in understanding the structure of the zonal winds in the deep interior of Jupiter.…”

Section: S U M M a Ry A N D R E M A R K Smentioning

confidence: 99%

“…In a recent study, Kong et al (2014) computed the gravitational signature produced by the Jovian equatorial winds that are equatorially symmetric and confined within a small equatorial region between the latitudes ϕ = ±25…”

mentioning

confidence: 99%

“…The study was motivated by the likelihood that the equatorial zonal jets cannot be strongly affected by either the Jovian magnetic field or the stable stratification (Liu, Goldreich, & Stevenson 2008;Lian & Showman 2010;Gastine & Wicht 2012). By comparing to the results from a model of the deep zonal winds at all latitudes, Kong et al (2014) found the equatorial zonal jets contribute 90 per cent of the high-order even gravitational coefficient J 12 in equation (1). Thus, the high-order gravitational coefficients -whose values were thought to reflect the penetration depth of the zonal winds -are nearly independent of the depth of the Jovian zonal winds in the non-equatorial regions with the latitudes 25…”

mentioning

confidence: 99%

See 1 more Smart Citation

Wind-induced odd gravitational harmonics of Jupiter

Monthly Notices of the Royal Astronomical Society: Letters

2015

While the rotational distortion of Jupiter makes a major contribution to its lowermost order even zonal gravitational coefficients J n with n ≥ 2, the component of the zonal winds with equatorial antisymmetry, if sufficiently deep, produces a gravitational signature contained in the odd zonal gravitational coefficients J n with n ≥ 3. Based on a non-spherical model of a polytropic Jupiter with index unity, we compute Jupiter's odd gravitational coefficients J 3 , J 5 , J 7 , . . . , J 11 induced by the equatorially antisymmetric zonal winds that are assumed to be deep. It is found that the lowermost odd gravitational coefficients J 3 , J 5 and J 7 are of the same order of magnitude with J 3 = −1.6562 × 10 −6 , J 5 = 1.5778 × 10 −6 and J 7 = −0.7432 × 10 −6 , and are within the accuracy of high-precision gravitational measurements to be carried out by the Juno spacecraft.Key words: planets and satellites: atmospheres -planets and satellites: interiors. I N T RO D U C T I O NJupiter is rotating rapidly, resulting in significant departure from spherical geometry: its shape eccentricity at the one-bar surface is E J = 0.3543 (Seidelmann et al. 2007). The shape and gravitational field of Jupiter can provide an important constraint on the physical and chemical properties of its interior. In 2016, the Juno spacecraft, now on its way to Jupiter, will make high-precision measurements of the Jovian gravitational field (Hubbard 1999; Bolton 2005) whose zonal external potential V g can be expanded in terms of the Legendre functions P n ,where M J is Jupiter's mass, n takes integer values, J 2 , J 3 , J 4 , J 5 , . . . , are the zonal gravitational coefficients, (r, θ, φ) are spherical polar coordinates with the corresponding unit vectors (r,θ,φ) and θ = 0 is the axis of rotation, R e is the equatorial radius of Jupiter and G is the universal gravitational constant (G = 6.673 84 × 10 −11 m 3 kg −1 s −2 ). At present, only the first three even zonal gravitational coefficients J 2 , J 4 , J 6 , which are mainly produced by the effect of rotational distortion of Jupiter, are accurately measured. By circling Jupiter in a polar orbit, the Juno spacecraft will carry out high-precision measurements of the gravitational coefficients up to J 12 (Bolton 2005). While both the rotational distortion and the equatorially symmetric zonal winds contribute to the even gravitational coefficients J n with n ≥ 2, the component of