  Recent precise measurements at Jupiter's and Saturn's gravity fields
constrain the properties of the zonal flows in the outer envelopes of these
planets. A simplified dynamic equation, sometimes called the thermal wind or
thermo-gravitational wind equation, establishes a link between zonal flows and
the related buoyancy perturbation, which in turn can be exploited to yield the
dynamic gravity perturbation. Whether or not the action of the dynamic gravity
perturbation needs to be explicitly included in this equation, an effect we
call the Dynamic Self Gravity (DSG), has been a matter of intense debate. We
show that, under reasonable assumptions, the equation can be solved (semi)
analytically. This allows us to quantify the impact of the DSG on each gravity
harmonic, practically independent of the zonal flow or the details of the
planetary interior model. The impact decreases with growing spherical harmonic
degree l. For degrees l=2 to about l=4, the DSG is a first order effect and
should be taken into account in any attempt of inverting gravity measurements
for zonal flow properties. For degrees of about l=5 to roughly l=10, the
relative impact of DSG is about 10% and thus seems worthwhile to include, in
particular since this comes at little extra costs with the method presented
here. For yet higher degrees, is seems questionable whether gravity
measurements or interior models will ever reach the required precision
equivalent of the DSG impact of only a few percent of less.
