External torque analysis

There will be many external torques affecting the satellite’s attitude and orbit such as the atmospheric drag, radiation pressure and gravity gradient. Of these, gravity gradient must be considered later on in the spacecraft’s development cycle as it depends on the craft’s internal mass layout. Torques produce an angular acceleration equal to:

\begin{equation} a=T/I \end{equation}

where a is acceleration, T is torque in newton meters and I is moment of inertial. For now we can assume our satellite will have an even mass distribution to give us a moment of inertia of 1/240 kg m^2

++Atmospheric Drag++
Even at such a high altitude, there is residual atmosphere that will produce a retarding drag force and torque proportional to the spacecraft’s speed squared.

\begin{align} F=\frac{1}{2}CArv^2 \end{align}

Where C is the drag coefficient, which describes the smoothness of the surface, r is the atmospheric density, and A is the cross sectional area. The rotational drag torque obeys a similar set of relations. The atmospheric density varies by several orders of magnitude as it is governed by solar weather, but it can be approximated using the exponential pressure relation, which gives us 1.963e-44 kg/m3 at 800km. This massive factor reduces the force to around 1e-36N. Clearly the main factor limiting the mission if it starts from this altitude is more likely to be radiation damage than orbital degradation. Similarly, the torque produced from atmospheric drag is going to be far too small to help detumble the satellite.

++Radiation Pressure++
Radiation pressure will affect the satellite as long as the sun is shining on it. It will produce a net torque if the craft has uneven coloration, a conservative oscillating torque that arises from the satellite's geometry as it spins, and a continuous force pushing the craft away from the sun.

The force generated by the radiation pressure on any side of the satellite facing the sun is in the order of 0.1nN. This will give us an effective course-modifying delta-v over a period of time, effectively 'squashing' our orbital trajectory against the earth on the sun-side, and causing greater eccentricity with an apoapsis on the night side of the earth. In one day, this will give us an orbital delta-v of 4.32 millimeters per second towards earth. Typical orbital velocities for low earth orbit

If the satellite has an uneven coloration such that one half of each face is completely reflective and the other completely absorptive, the imbalance of torque will be 2.5pNm. To find the net angular velocity increase per day, we integrate over 24hrs (that's 86400 seconds) to give us 51.8 microradians per second per day. That's 18.9 milliradians per second for an entire year, very low.

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