To keep the attitude of a spacecraft stable, satellites or space probes are often put
into a rapid spin, either permanently or temporarily, e.g. during a long coasting period.
This works because due to the conservation of angular momentum the orientation of the
spin axis of such a system does not change in the absence of any massive external influence.
If the transmitter antenna does not lie on the rotation axis, it will trace a circular path
around the axis during one rotation. If the axis is perpendicular to the line-of-sight,
the antenna will approach us during one part, but recede from us in the other. Consequently,
the frequency of the emitted radio signal will be received on a slighly higher frequency, and
then in a lower frequency. The Doppler shift of the signal follows a sine law:
Δf (t) = Δfmax sin (2 π t/T)
with the duration T for one rotation. The maximum frequency shift
Δfmax = -v/c * f = -(2π d/T) / c *f
depends on the antenna's speed v on its circular path, the distance d between the antenna and
the centre of the craft's mass (hence of rotation), the speed of light c, and the transmission
frequency f.
When the rotation axis is parallel to the line-of-sight, the antenna does not change its
distance to the observer, and the signal does not experience any Doppler shift.
For any angle α between the plane of rotation and the line-of-sight:
we get this simple expression
Δfmax = -(2π d/T) / c * f * sin(α)
When we observe the time variation of the received frequency, we can draw important information:
A very nice example is the Ikaros space craft, which on June 4, 2010 for the first time tested the deployment in space of a solar sail. The measurements by F5PL before the unfolding:
and just after the deployment
showed not only that the rotation was substantially slowed down - due to the conservation of angular
momentum - by the larger extent of the space craft, but also that the observed Doppler shift amplitude
can be well explained with the presumed position of the transmitter antenna at the outer edge
of the craft's body (d = 80 cm) and an angle α = 40º:
The rotation rate of 25 rpm means a period of 2.4 sec. During one rotation the antenna covers a circular
path of 5 m, with a speed of 2 m/s. At f=8.4 GHz this corresponds to a frequency shift of 59 Hz.
Hence the angle should be near 40º.
Ikaros was observed from DL0SHF all the time before the sail deployment until about one month later. A selection of our records shows the slowing down of the rotation and the decrease of the Doppler amplitude:
All data can be presented in this form (the green arrow marks the sail deployment):