Determination of the time dependency (secular variation) of a planet’s magnetic field provides a window into understanding the dynamo responsible for generating its field. However, of the six Solar System planets with active dynamos, secular variation has been firmly established only for Earth. Here, we compare magnetic field observations of Jupiter from the Pioneer 10 and 11, Voyager 1 and Ulysses spacecraft (acquired 1973–1992) with a new Juno reference model (JRM09). We find a consistent, systematic change in Jupiter’s field over this 45-year time span, which cannot be explained by changes in the magnetospheric field or by changing the assumed rotation rate of Jupiter. Through a simplified forward model, we find that the inferred change in the field is consistent with advection of the field by Jupiter’s zonal winds, projected down to 93–95% of Jupiter’s radius (where the electrical conductivity of the hydrogen envelope becomes sufficient to advect the field). This result demonstrates that zonal wind interactions with Jupiter’s magnetic field are important and lends independent support to atmospheric and gravitational-field determinations of the profile of Jupiter’s deep winds

Bloxham, J.

Bolton, S. J.

Cao, H.

Connerney, J. E. P.

Moore, K. M.

Stevenson, D. J.

Caltech Authors - Main

Letters
https://doi.org/10.1038/s41550-019-0772-5
Time variation of Jupiter’s internal magnetic field 
consistent with zonal wind advection
K. M. Moore   1*, H. Cao   1,2, J. Bloxham   1, D. J. Stevenson   2, J. E. P. Connerney3,4 and S. J. Bolton5
1Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA, USA. 2Division of Geological and Planetary Sciences, California Institute 
of Technology, Pasadena, CA, USA. 3Space Research Corporation, Annapolis, MD, USA. 4NASA Goddard Space Flight Center, Greenbelt, MD, USA. 
5Southwest Research Institute, San Antonio, TX, USA. *e-mail: kimberlymoore@g.harvard.edu
SUPPLEMENTARY INFORMATION
In the format provided by the authors and unedited.
NaTurE aSTroNoMy | www.nature.com/natureastronomy
 
 
1 
 
Supporting Information 
Time-variation of Jupiter's internal magnetic field consistent with zonal 
wind advection 
K. M. Moore1, H. Cao1, J. Bloxham1, D. J. Stevenson2, J. E. P. Connerney3,4, and S. J. Bolton5 
1Department of Earth & Planetary Sciences, Harvard University, Cambridge MA 02138, USA 
2Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA 
3NASA Goddard Spaceflight Center, Greenbelt, MD 20771, USA 
4Space Research Corporation, Annapolis, MD 21403, USA 
5Southwest Research Institute, San Antonio, TX 78238, USA 
 
 
Introduction  
This supporting information contains additional details on the modeling performed in the main 
text, and spherical harmonic coefficients for our zonal wind advection (ZWA) model. Specifically, 
we enclose a figure (Supplementary Figure 1) demonstrating our external field models and fit to 
the data, and the parameters used therein (Supplementary Table 1); a figure showing the effect 
of alternative rotation periods on the enclosed secular variation (Supplementary Figure 2); a 
comparison of Jovian rotation periods between several published studies (Supplementary Table 
2); a figure showing the fit of our ZWA model to the inferred secular variation for an alternative 
rotation period (Supplementary Figure 3); a figure showing Jupiter’s zonal winds projected to 
alternative projection depths (Supplementary Figure 4); a figure showing the fit to the inferred 
secular variation using alternative wind projection depths (Supplementary Figure 5); and a figure 
showing the relative magnitude of each error source for the past spacecraft magnetometer data 
used in our study (Supplementary Figure 6).  
We also enclose a list of spherical harmonic coefficients for our ZWA (zonal wind advection) 
model of Jupiter’s secular variation using zonal winds projected to 0.95RJ (Supplementary 
Dataset 1), 0.94 RJ (Supplementary Dataset 2), and 0.93 RJ (Supplementary Dataset 3). These 
datasets are in ASCII format. 
 
 
 
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 Supplementary Figure 1. Fitting the magnetodisk model. This plot compares the fit of our 
magnetodisk models (red, calculated based off of Connerney et al., 1981) to the observed 
magnetic field data minus JRM09 (Connerney et al., 2018; black).   
 
 
 
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Supplementary Figure 2. Comparing alternative rotation periods. We show how the inferred 
secular variation depends on Jupiter’s rotation period. We show the change in magnetic field (y-
axis) between Juno and past spacecraft missions along the spacecraft flight path (x-axis), for 
the r-, theta-, and phi-components (top, middle, and bottom respectively). We plot: the slowest 
and fastest rotation periods within the System III (1965) uncertainty bounds (respectively Ψ = -
0.36 deg/yr, magenta; and Ψ = +0.36 deg/yr, red); the standard System III (1965) rotation period 
(Ψ = 0, black); and the rotation period that minimizes the inferred secular variation across the 
four missions (Ψ = +0.094 deg/yr, cyan). Since no rotation period reduces Delta B to zero 
(within data error bounds) for all missions, a rotation period is not enough to explain the inferred 
secular variation in Jupiter’s magnetic field.  
 
 
 
 
 
 
 
 
 
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Supplementary Figure 3. Models of Jupiter’s magnetic secular variation using an 
alternative Jovian rotation period We compare models of Jupiter’s secular variation (JSV 
[Ridley & Holme, 2016] in blue, and ZWA [this study] in magenta), to the actual inferred change 
in Jupiter’s internal magnetic field between Juno and past spacecraft missions (black), where 
the coordinates have been adjusted using a coordinate system drift rate of Ψ = 0.094 deg/yr 
(see Methods for details). The shaded grey region denotes the measurement error bars (see 
Methods for calculation details). For this figure, we use a magnetic drift rate of 0.025 m/s for the 
ZWA model, instead of 0.0375 m/s. This is because some of the apparent drift of magnetic 
features could be accommodated by a rotation period change (longitude shift) in the coordinate 
system, instead of advection by zonal winds.  
 
 
 
 
 
 
 
 
 
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Supplementary Figure 4. Zonal wind geometry using alternative projection depths Here 
we show the wind geometries that result when projecting Jupiter’s surface winds to depths of 
0.93 (green), 0.94 (orange), and 0.95 RJ (blue), and scaling the peak amplitude to 1.8 cm/s, 2.1 
cm/s, and 3.75 cm/s respectively.  
 
 
 
 
 
 
 
 
 
 
 
 
 
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Supplementary Figure 5. Comparing alternative wind depths Here we show the fit of our 
ZWA model to the inferred change in Jupiter’s internal magnetic field between Juno and past 
spacecraft missions (black). The shaded grey region denotes the measurement error bars, 
shown here bracketing the inferred secular variation (see Methods in main text for calculation 
details). We show ZWA models calculated by projecting Jupiter’s surface winds to three 
different depths (see Figure S4): 0.93 RJ (cyan), 0.94 RJ (red), and 0.95 RJ (royal blue). The 
0.93 RJ and 0.94 RJ models plot nearly on top of each other. The 0.95 RJ model shown here is 
the same ZWA model that is shown in the main text.  
 
 
 
 
 
 
 
 
 
 
 
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Supplementary Figure 6. Comparison of magnetometer data error magnitudes by source 
for past spacecraft missions Here we show the relative contribution of each error source 
(baseline sensor accuracy in magenta, data quantization error in red, and pointing error in blue) 
to the total error source (black line) along the spacecraft flight track for Pioneer 10, Pioneer 11, 
Voyager 1, and Ulysses.  
We assume the pointing error is an arbitrary rotation about the spacecraft’s presumed 
orientation and take an order of magnitude estimate, rather than modeling it component by 
component. In reality, the errors are likely anisotropic. For example, consider the case of a 
magnetic field B = (0,0,B) measured in the spacecraft frame (x,y,z). If the assumed attitude was 
wrong by a small rotation angle α relative to the z axis, then the field transform between 
spacecraft coordinates and a reference coordinate system (such as System III) will lead to 
errors of (1-cos(α))B in Bz, and sin(α)B in the field perpendicular to the z axis (approximately -
0.5α2B and α respectively for small angles). See Holme & Bloxham (1996) for a full treatment of 
anisotropic attitude errors in spacecraft data analysis.  
 
 
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 Supplementary Table 1. Magnetodisk parameters from Monte Carlo analysis.  
Parameter Value 
 Pioneer 10 Pioneer 11 Voyager 1 Ulysses 
R0 7.22 RJ  3.04 RJ 7.65 RJ 7.66 RJ 
D 3.47 RJ 2.82 RJ 2.52 RJ 3.42 RJ 
µ0I 282.0 367.7 345.8 292.2 
θ 9.6 deg 9.7 deg 5.3 deg 7.7 deg 
φ 254.1 deg 254.1 deg 208.2 deg 283.7 deg 
This table shows the magnetodisk parameters for our ZWA model using winds projected to 
0.95RJ. Here θ is the dipole tilt, and φ is the dipole londitude. R1 was kept constant at 50 RJ 
(Jupiter radii). These parameters assume the standard System III (1965) Jovian rotation period, 
and correspond to the 6-parameter model described in Connerney et al. (1981).  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Supplementary Table 2. Comparison of Jovian rotation rate studies 
Reference Rotation period 
Coordinate system drift rate, Ψ (deg/yr) 
System III (1965) 9h 55min 29.711 +/- 0.04 sec N/A 
Higgins et al. (1997) 9h 55min 29.6854 +/- 0.0035 sec -0.23 
Russell et al. (2001) 9h 55min 29.710 sec -0.0089 
Ridley & Holme (2016) 9h 55min 29.7258 sec +0.13 
This study:   
Pioneer 10 optimal fit 9h 55min 29.718 sec +0.069 
 Pioneer 11 optimal fit 9h 55min 29.722  sec +0.096 
Voyager 1 optimal fit 9h 55min 29.723  sec +0.11 
Ulysses optimal fit 9h 55min 29.723  sec +0.11 
 
We convert the rotation periods published in other studies to a variety of units for ease of 
comparison. The drift rates (Psi) are given with reference to 2017.0: 
φ’ = φ + Ψ*(2017-t) 
Where φ’ is the rotated E longitude in the new coordinate system, φ is the original datapoint E 
longitude, Ψ is the coordinate system drift rate in deg/yr, and t is the datapoint decimal year.  
 
With this set of definitions, 2017.0 is taken as the reference date for our rotated coordinate 
systems, and has an equivalent longitude to System III (1965). Positive drift rates indicate 
increasing East longitudes before 2017, and decreasing E longitudes after 2017 (a slower 
rotation period), and negative drift rates indicate a faster rotation period. Coordinates at 2017.0 
will be unchanged. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Supplementary Dataset 1. This dataset contains spherical harmonic coefficients for our zonal 
wind advection (ZWA) model using winds projected to 0.95 RJ, in nT/yr. We use RJ = 71492km 
and a drift rate of 3.75 cm/s. To calculate the coefficients for a different field drift rate, simply 
multiply them by a scalar (e.g. multiply current values by 2 for a drift rate of 7.5 cm/s). The 
coefficients appear in the following order: g10, g11, h11, g20, g21, h21, g22, etc. 
 
Supplementary Dataset 2. This dataset contains spherical harmonic coefficients for our 
zonal wind advection (ZWA) model using winds projected to 0.94 RJ, in nT/yr. We use RJ = 
71492km and a drift rate of 2.1 cm/s. To calculate the coefficients for a different field drift rate, 
simply multiply them by a scalar (e.g. multiply current values by 2 for a drift rate of 4.2 cm/s). 
The coefficients appear in the following order: g10, g11, h11, g20, g21, h21, etc. 
 
Supplementary Dataset 3. This dataset contains spherical harmonic coefficients for our zonal 
wind advection (ZWA) model using winds projected to 0.93 RJ, in nT/yr. We use RJ = 71492km 
and a drift rate of 1.8 cm/s. To calculate the coefficients for a different field drift rate, simply 
multiply them by a scalar (e.g. multiply current values by 2 for a drift rate of 3.6 cm/s). The 
coefficients appear in the following order: g10, g11, h11, g20, g21, h21, g22, etc. 
 
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Edwards, T. M., Bunce, E. J., and Cowley, S. W. H. (2001). A note on the vector potential of 
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