Would magnetic torquers placed in a polar satellite keep the satellite stable?
If it would work, would it be very efficient?
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1$\begingroup$ I assume you're wondering about this because of the structure of the Earth's magnetic field (i.e. magnetic poles near the geographic poles)? If so, please make this more clear in your question. $\endgroup$– user29Aug 26, 2013 at 14:42
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$\begingroup$ @Hash Do you mean magnetorquers (as active attitude control), as pointed out by SF, or permanent magnets (as passive attitude stabilization)? Your question is slightly ambiguous. $\endgroup$– s-m-eAug 26, 2013 at 14:57
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2$\begingroup$ @ernestopheles: Magnetorquers are active control, and this is what Hash is asking about. $\endgroup$– Deer HunterAug 26, 2013 at 15:13
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$\begingroup$ @DeerHunter I do not see him mentioning 'active'. Polar orbits are technically no issue for magnetorquers. However, they become an issue if you intend to passively stabilize your satellite with magnets (which is at least being attempted). I have seen too many misconceptions in this field, so that's why I would like Hash to confirm, what he is asking about. $\endgroup$– s-m-eAug 26, 2013 at 15:17
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$\begingroup$ Since the very term "magnetorquers" is both synonymous with and a portmanteau of "magnetic torquers", I think DeerHunter's is a safe assumption, and I do not think the question as worded is ambiguous. I have personally never seen the term used to describe passive magnetic stabilization. Just my 2 cents. $\endgroup$– user29Aug 26, 2013 at 15:24
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Yes, and in fact, magnetic torquers are more efficient in polar orbits than in equatorial ones. This makes sense if you look at a picture of Earth's magnetic field lines:
When a spacecraft is in a polar orbit, it experiences higher magnetic flux (passes through more lines), and thus is exposed to a stronger magnetic field.
The map below shows a map of the Earth's magnetic field strength (numbers are in nanoTeslas).
As you can see (and as this page confirms), the magnetic field strength at the poles is about twice that of the equatorial field strength.
Now, how does this apply to magnetic torquers on a spacecraft? Well, the torque $T$ produced by a magnetic torquer depends on the magnetic field strength $B$ and the dipole induced by the torquer $D$ (which is proportional to the current applied to the torquer): $$T=DB$$ As you can plainly see, the higher the magnetic field strength ($B$), the larger the torque ($T$).
Going a step further, altitude is much more of a factor. $B$ can be approximated by: $$B=2M/R^3$$
where $M$ is the magnetic moment of the Earth (about $7.96 \times 10^{15} \text{tesla} \cdot \text{m}^3$) and $R$ is the distance from Earth's dipole center and the spacecraft. As you can see here, higher altitudes decrease the field strength drastically, which in turn decreases the torque you'll see from your magnetic torquer. This is why you only see magnetic torquers on LEO spacecraft.
Note: for the last section I used Space Mission Analysis and Design (3rd ed., Wertz, J.R. and Wiley, J.L., editors, 1999) as a reference.