# Theory of Rotational Behavior of a Rocket Body

After a third stage has inserted a payload into orbit, it is disconnected from the payload by explosive bolts. The third stage, usually called the rocket, is no longer stabilized after this phase. So, it will be in a state of free rotational motion in its orbit around the Earth. For cylindrical objects (like rockets), this motion consists of a spin (about the axis of symmetry) combined with a precession of that axis about a fixed direction, namely the direction of the angular momentum of the rotation.

This free or natural precession should not be confused with the well known precession of the Earth's axis. On the other hand, it is true that due to the Earth's gravity the angular momentum vector of the rocket will precess around the vector normal to the orbital plane.

The spin-frequency and the angle between the axis of symmetry and the angular momentum are defined by the initial conditions, such as the force with which the rocket was disconnected from the payload.

Flashing rockets usually show a regular brightness pattern. From this we can probably deduce that they tumble about the short body-axis, which in this case coincides with the angular momentum vector ( ). Concluding we can say that some force changes the original rotation quickly into a tumbling motion. Additionally, it appears that the flash period usually increases with time.

This is caused by the force that the Earth's magnetic field exerts on the rocket. Other forces, such as the Earth's gravity and aerodynamic friction are negligible for most rockets. Rockets in orbits with heights lower than 200 km are influenced more by aerodynamic than by magnetic forces. But those rockets will not survive very long in orbit, since they quickly decay due to atmospheric drag.

The Earth's magnetic field induces an electric field in the tumbling rocket, which usually consists of a conducting material. This electric field causes a current to flow in the rocket. The Earth's magnetic field exerts a force on this current, the Lorentz force. It is this force, more exactly the torque associated with this force, that dominates the rotational state of the rocket.

Firstly, the torque will cause a long rocket to tumble around a short body-axis, according to this formula : with : the initial value of , just after launch (t=0) the relative magnetic field strength (see below)

Q Q is expressed in SI units. Q is a positive constant, depending on the construction of the rocket such as radius r, length l, conductivity for Aluminum), thickness of rocket's skin D (typically 1 mm), moment of inertia around the short body axis (typically ); the strength of the Earth's magnetic dipole and the semi-major axis a of the orbit. The semi-major axis is the mean height for a circular orbit, which we assume anyway. This assumption is usually a very good one.

w  is the angle between the angular momentum and the local magnetic field of the Earth.

K is constant and is the ratio between the moment of inertia around the short axis and around the long axis. K is a measure of the oblateness of the rocket. It is equal to 0.5 for a short flat disc, equal to 1 for a sphere and bigger than 1 for a 'long' rocket. For a typical Zenit or Cosmos rocket it is 3.67. A 'flat' satellite has a K-value smaller than 1, so w can change sign. This means that the rotation axis will not necessarily asymptotically go towards a constant attitude with respect to the satellite, if is variable. Since payloads will not necessarily be in a state of tumble, the results of Chapter 6 (for objects that are not in a state of tumble) apply. This is why payloads (mostly 'flat') usually have a rather complex flash pattern. Another reason is their complex shape which is due to antennae, booms and solar panels.

Apart from the influence on the rotation angle , the magnetic torque will also cause the rotation period to increase exponentially : This formula is strictly speaking only valid for . The characteristic time of this increase ( ) i.e. the time in which the rotation period increases with a factor of e (=2.71828...), is at least K times bigger than the characteristic time of the motion for long rockets. This is why we observe a reasonably regular flash pattern already shortly after launch.

The local magnetic field of the Earth changes as the rocket moves along its orbit, hence and also change. Simply substituting these local and instantaneous values by their averages over one orbit solves this problem, since we are interested in long-term changes of the flash period. To take into account the fact that the geographic north pole does not coincide with the magnetic pole of the Earth, we also average over one day. We then find the following expression for the magnetic field strength : with :

i the inclination of the orbit the geographic latitude of the Earth's magnetic pole ( degrees).

The last term in this equation is only a small correction to the first ones. Hence, rockets with higher inclination will experience a higher magnetic field strength, and the magnetic friction will be larger. So, the increase of the rotation period will be faster, and the characteristic time is smaller.

The value of depends on the orientation of the angular momentum. This means that similar rockets in identical orbits will not necessarily have the same characteristic time. For orbits with an inclination between 60 and 120 degrees, the difference can be up to a factor of 2.

Exact theoretical values of the characteristic time are difficult to calculate, since the material constants on which Q depends are not exactly known. This is why we calculated the dependency (on i) of the extreme and average values of The penultimate column of table 5 contains the declination of the angular momentum for which V is minimal. In that case the angular momentum is perpendicular to the line of nodes of the satellite's orbit.

The last column gives the maximal value of K for which it is possible that w becomes negative, so that does not necessarily asymptotically go to 90. If the satellite asymptotically goes into a tumble.

It is clear from Table 5 that satellites in a polar orbit will lose their rotational energy at a higher rate (i.e. become steady) than satellites in an equatorial orbit.

Table 5 : Dependency of h, V, and on the inclination The magnetic torque does more than influence the rotation period. The torque causes the angular momentum to move towards the orientation where V is minimal. Due to this, the gravitational precession of the angular momentum and also the fact that the line of nodes rotates (due to the Earth's oblateness), the average value of will vary over a longer period. As a consequence the increase of the rotation period is not purely exponential, but could show a 'wavy' character (i.e. the characteristic time is not a constant, but varies with time). The amplitude of this variability of the characteristic time will not exceed the factor two mentioned earlier for high-inclination satellites.

The gravitational precession of the angular momentum and the rotation of the line of nodes also have as a consequence that the angular momentum can have a fixed direction only for equatorial or polar satellites. In this case the angular momentum vector is parallel to the Earth's rotation axis.

The theory as explained above can only explain an increase of the rotation period, not a decrease, i.e. an acceleration. There are however numerous examples of accelerations in the PPAS.

A possible explanation for this behavior is that the acceleration is caused by a fuel leak. There is always a little bit of fuel left after the orbital insertion. If the remaining fuel leaks out of the rocket, a force is exerted on the rocket body. This can cause a spinning up or down. This behavior has been observed with several third stages shortly after launch. Other rockets start accelerating a few years after launch. This time-delay may be due to the fact that the (usually) corrosive fuel has to eat its way through the hull before it can escape. This process takes time, hence the delay. Another possible cause for a fuel leak is a collision with a micro-meteoroid or space debris. This collision in itself can cause a very sudden acceleration. This also has been observed on some rockets.

Finally, some rockets have undergone multiple accelerations over the years. Though a theory is not yet available it is thought that this may be due to a fuel freezing-evaporation cycle. When the satellite is in the sun for a longer period, the fuel starts to evaporate and can escape. As soon as the rocket enters the shadow of the Earth for longer times, the fuel freezes and can no longer escape.

The next section deals with the application of this theory to a set of observations.

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