1
$\begingroup$

Can someone please provides me with the recursion relations of Associated Legendre Polynomials when using Schmidt quasi-normalization? I need that in the context of Geomagnetism to obtain the Spherical Harmonics coefficients g and h.

$\endgroup$
2
$\begingroup$

FYI, the usual reference relied on for the equations used in geomagnetism is:

  • Langel, R. A. "Chapter four: Main field." Geomagnetism, edited by JA Jacobs (1987).

If it wasn't for lockdown I'd have copied from the book locked in my office to make sure I got it right...but until someone corrects me...

The Schmidt quasi/semi-normalisations for the associated Legendre polynomials are given by these recursions, in the form you'd actually use for calculations typically:

Note that if you use Schmidt normalisation of Legendre polynomials from a given software language to do calculations, you do not want to include the Condon-Shortley phase factor of . So check if it is included in the Legendre polynomial and/or normalisation formula.

Edit: Some links to open source geomagnetism compatible software for calculating Schmidt normalised associated Legendre polynomials.

Python: ChaosMagPy

Python/Fortran-95: SHTools

| improve this answer | |
$\endgroup$
  • $\begingroup$ That exactly what I was looking for, except I couldn't find the mentioned reference anywhere on the web, so thanks for the hint. However, I really need a confirmation about the formula correctness including the derivatives too. $\endgroup$ – Ayoub Aug 24 at 16:06
  • $\begingroup$ I don't think there is a digital version of Langel (1987) anywhere online, but I just found I could view enough of the following reference through Google Books to confirm the equations: Wertz, J. R. "Spacecraft Attitude Determination and Control", (1978). Use the "Search inside" option and search for "Appendix H", you can see the first few pages about Schmidt normalisation. $\endgroup$ – WJB Aug 24 at 16:27
  • $\begingroup$ That was useful, thanks again! So, the factors S are used with Gauss functions to obtain Schmidt functions, where: P_Schmidt = S * P_Gauss. But, it's still a long way to go with. I would love to compute directly the P_Schmidt functions since we know that there's recursion relations for them out there. $\endgroup$ – Ayoub Aug 24 at 17:33
  • $\begingroup$ Are you actually after the final recursive formula for the Schmidt P_n^m(cos(theta)) functions themselves, or are you ultimately trying to write code to calculate these values for geomag problems? I can add links to open source code in a few languages much more easily than I can work back and type up the formula working from those codes right now! $\endgroup$ – WJB Aug 24 at 19:15
  • $\begingroup$ Exactly I'm trying to implement a code to calculate those values, any open source code is welcomed! but I can't just take the code without the formula since I need to document that. $\endgroup$ – Ayoub Aug 24 at 21:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.