The Winds of Saturn and their effect on its figure and rotation rate

by Daanish Maqbool

TERPS Conference, 2012

College Park, MD

Introduction

The rotation rate of Saturn has been the subject of some controversy1,2. The rotation rate inferred from radio emissions (e.g. Saturn Kilometric Radiation, SKR) has been observed to change over time1. Using Voyager (1980 - 1981) data, the period was observed to be 10 hours, 39 minutes, 24 seconds, but data from Cassini (2002 - 2004) showed a period of 10 hours, 45 minutes, and 45 seconds. For these reasons, Anderson and Schubert2 have attempted to derive the rotation rate of Saturn using a novel method which is independent of any radio emissions data. This is accomplished by using the measured gravitational properties of Saturn, in addition to zonal wind velocities calculated by tracking cloud features in images supplied by Voyager 23.

Calculation Method

The first step in the process is to calculate a reference geoid for Saturn, i.e. a surface of equal gravitational potential at a point that is co-rotating with the surface of Saturn. In polar coordinates, this surface is defined as4

( ) ( ) 2221

22 cos21

sin, rPJr

GMr

GMrU

iii +=

=

Where P2i is the Legendre polynomial of degree 2i and is the assumed rotation rate. The gravitational coefficients J2i can be obtained from Voyager and/or Cassini radio Doppler data2. In practice, the preceding equation must be iterated for r at every latitude () for the reference value of the gravitational potential, Uref, which was set to the value calculated at the polar radius, (because the polar radius is known from radio occultation data4).

Once a reference geoid of Saturn has been calculated, the next step is to calculate how the winds would perturb this geoid. The change in height of the geoid (i.e. height above and normal to the reference geoid) resulting from the winds is4

( ) ( ) ( )( )

( ) ( )

pi

dr

r

VVg

h refref

ref

ref

WW

cos

sincos

212 +

+=

where VW is the zonal wind velocity and is the average gravitational acceleration along the height of the geoid. ref is defined as ( )rref gg arctan=

where

( ) ( ) ( )

( )[ ]

sin132

sin12,

22

12

2

222

Pr

Pr

RJir

GMr

GMrg

ii

i

ir

+

++=

=

( ) ( ) ( )

d

dPr

ddP

r

RJr

GMrg

i

ii

isin

31sin

,22

1

22

22

=

=

For an assumed rotation rate, , a certain height distribution above the reference geoid will be produced. Anderson and Schubert2 calculated the value of for which the change in height of the geoid due to the winds in minimized in an average sense (Fig. 1). This corresponds to the condition that would minimize the energy needed to drive the winds2. Their value of which minimizes height variations corresponds to a rotation rate of 10 hours, 32 minutes, and 35 seconds.

Mismatch at the Poles

A striking feature of figure 1 is that the poles are offset by about 10 km, which is a significant discrepancy and has been described by the authors as puzzling. This implies that the centers of figure and mass of Saturn are not coincident2. While there is no clear reason for such an offset (it has been speculated that this is some seasonal

Figure 1 Height due to winds above the reference geoid for different values of rotation rate (PIII is slowest rotation and PI is the fastest). Adapted from ref. 2.

effect4), I will explore the possibility that this is due to uncertainties in the zonal wind data by repeating the calculations of refs. 2 and 4, but with slightly modified data sets.

Correction 1: Zero Velocity at Poles

The original zonal wind data provided by Smith et al.3 is in the form of discrete points, whereas the results of refs. 2 and 4 are in the form of continuous lines. One is therefore led to believe that some aspect of curve-fitting was involved in the process of calculating geoid heights. Furthermore, the zonal wind velocity at the poles in not specified in the data of Smith et al.3 (which is expected to be zero), and it is therefore possible that the curve-fitting processes of refs. 2 and 4 produced a non-zero velocity at the poles which may be reason for the discrepancy in the pole heights. After adding zero velocity data points at the poles to the original data set of Smith et al.3 (and using linear interpolation to calculate values in between given data), the calculations for geoid height were performed and the results are plotted in figure 2.

-50 0 50 100 150-100

-50

0

50

100

Altitude (km)

(La

titude

)

Height due to w indsReference Geoid

It can be seen from figure 2 that the wind velocities at the poles are not the reason for the pole height mismatch. After inspecting the integral in the equation for h(), it is apparent that the asymmetry in the wind velocities is responsible for this discrepancy.

Correction 2: Wind Velocity Magnitude

The effects of magnitude uncertainties in the wind velocity data were also explored. The velocities were perturbed by + 20 m/s, and the resulting heights were calculated (Fig. 3). It can be seen that the magnitude of the heights changes, but there is no appreciable change at the poles.

0 50 100 150

-50

0

50

Altitude (km)

(La

titude

)

VW - 20 m/sVWVW + 20 m/s

Correction 3: Wind Velocity Latitude

For the final correction, the data was shifted in latitude. It was found that the pole heights were sensitive to this latitude shift, and a shift in the data of -2.5 removed the pole height mismatch (Fig. 4).

0 50 100

-50

0

50

Altitude (km)

(La

titude

)

- 2.5 deg.data4

The uncertainty in the data of Smith et al.3 has not been specified by the authors. However, this study suggests that if the uncertainty in the latitude of the zonal wind data3 is up to 2.5, then the pole height mismatch can be accounted for.

References

1Gurnett, D. A. et al., The variable rotation period of the inner region of Saturn's plasma disk, Science, 316, 20th April, 2007 2Anderson, J. D. and Schubert, G., Saturns Gravitational Field, Internal Rotation, And Interior Structure, Science, 317, 7th September, 2007 3Smith, B. A. et al., A New Look at the Saturn System: The Voyager 2 Images, Science, Vol. 215, 29th January, 1982 4Lindal, G. F., Sweetnam, D. N., and Eshleman, V. R., The Atmosphere of Saturn: An Analysis of the Voyager Radio Occultation Measurements, Astronomical Journal, Vol. 90, no.6, June, 1985

Figure 2 Height variations above the reference geoid with zero wind velocity at the poles.

Figure 3 The effect of zonal wind velocity magnitude on geoid height

Figure 4 The effect of a shift in wind location (latitude). A downward shift of 2.5 removes the pole height mismatch