Question 5 sun earch doc check RESOLVED

How did I change reference frames in sun earth doc? PhL 2.28.17 This is a due diligence test that luckily passed OK. I just corrected a bad frame change in frames doc Section 8.8 regarding moon earth. I am now concerned that I might have made this same error elsewhere in my docs. In particular, I am worried about sun earth doc (astronomy doc) where I shift from north pole to local location on the earth. How did I do that frame change in that doc ??? (very nervous right now. ...) Here is one set of equations The black equatorial circle is associated with Frame S with x,y,z axes, while the ecliptic red circle is associated with Frame S' with x',y',z' axes. Staring at the figure one sees that ' = Rx(θtmax) and ' = => x' = x ' = Rx(θtmax) (6.1) where θtmax = 23o. According to the rule that vectors rotate backwards from the unit vectors (see Appendix B), we then have in 2D that r = Rx(θtmax)r' => = => Notice the quote "according to the rule that vectors rotate backwards from the unit vectors". So at least I was aware of this fact at the time, so the last equation is then compatible with the unit vector equations above it. (whew!). I make reference to Appendix B which is my sun-earth take on active versus passive. I will read it right now. // It is just fine! It does not need notation like (V)' because in that doc I don't have vectors like v' which have other definitions. All OK at least at this point in sun earth. At this point in sun-earth I am using Frame S = celestial sphere and Frame S' = tilted ecliptic as shown in Fig 6.1. The next Big Deal is Section 7 where I have Frame S = celestial sphere and Frame S' = local system on surface of the earth. So how do I accomplish this transformation? I first do Cartesian to Cartesian in Section 7 (c) based on this equation = cosφsinθ + cosφcosθ - sinφ = sinφsinθ + sinφcosθ + cosφ = cosθ – sinθ . (7.10) which I got from a sun-earth appendix. I sure hope this agrees with frames doc on the same thing: = sinθcosφ + cosθcosφ - sinφ = sinθsinφ + cosθsinφ + cosφ = cosθ - sinθ . (E.2.7) OK that is a very good thing! They agree. Look now at sun earth app A where this is derived. Ouch, I quote it from another of my docs, which is Ref [6] of sun earth which is frames doc!!! But I probably have changed eq nums? I refer to this as Appendix A (A.13.c) of frames doc. YES! That is still correct, I have not changed it! I have this in both App A and App E of frames doc. I am now reading near (7.10) of sun earth doc. The vector b is earth center to local frame origin on surface of the earth in Fig 7.1, same as in frames doc. I am asking about the sun position in Frame S'. In (7.1) I have = e'3 = - e'2 = e'1 , (7.1) so these are how the Frame S basis vectors like are related to the Frame S' ones like e'3 in Fig 7.1. I then install these into the triple of equations shown above as E.2.7 to get = cosφb sinθ1 e'3 – cosφb cosθ1 e'2 – sinφb e'1 = sinφb sinθ1 e'3 – sinφb cosθ1 e'2 + cosφb e'1 = cosθ1 e'3 + sinθ1 e'2 . (7.11) And then I write r' ≈ r = x + y + z = x (cosφbsinθ1 e'3 – cosφbcosθ1 e'2 – sinφb e'1) + y (sinφbsinθ1 e'3 – sinφbcosθ1 e'2 + cosφb e'1) + z (cosθ1 e'3 + sinθ1 e'2) = [– x sinφb + y cosφb]e'1 + [– x cosφbcosθ1 – ysinφbcosθ1 + z sinθ1]e'2 + [ x cosφbsinθ1 + y sinφbsinθ1 + z cosθ1]e'3 (7.12) = x' e'1 + y' e'2 + z' e'3 so that the Cartesian sun location coordinates in Frame S' are given by x' = – x sinφb + y cosφb y' = – x cosφbcosθ1 – ysinφbcosθ1 + z sinθ1 z' = x cosφbsinθ1 + y sinφbsinθ1+ z cosθ1 . (7.13) This is "basis vector method" and the result looks good to me. I now have local Cartesian coordinates of the location of the sun. Here (x,y,z) is r in Frame S which is the center of earth aligned frame. I know that the sun's path in this Frame S is given by x = R cosθt cosφ y = R cosθt sinφ z = R sinθt (7.6) and I then get x' = R cosθt sin(φ –φb) y' = R [sinθt sinθ1 – cosθt cosθ1 cos(φ–φb) ] z' = R [sinθt cosθ1 + cosθt sinθ1 cos(φ-φb)] (7.14) So these are the sun coordinates in local earth Frame S'. So this concludes my Cartesian → Cartesian transformation, it looks OK. Then in Section 7 (d) I convert to θ' and φ' as Frame S' spherical coordinates! x' = r'sinθ'cosφ' y' = r'sinθ'sinφ' z' = r'cosθ' cosθ' = [sinθt cosθ1 + cosθt sinθ1 cos(φ-φb)] sinθ' = ρ'/r' = (7.27) cosφ' = x'/ρ' = [cosθt sin(φ –φb)] / sinφ' = y'/ρ' = [sinθt sinθ1 – cosθt cosθ1 cos(φ –φb) ]/ tanφ' = y'/x' = [sinθt sinθ1 – cosθt cosθ1 cos(φ –φb) ]/ [cosθt sin(φ –φb)] (7.28) Therefore, we may write θ' = cos-1[sinθt cosθ1 + cosθt sinθ1 cos(φ-φb)] φ' = tan-1 ( [sinθt sinθ1 – cosθt cosθ1 cos(φ –φb) ] / [cosθt sin(φ –φb)] ) . (7.29) OK enough. I think I can rest easy that I did not do a major screw-up in sun earth on this subject. As noted, I was aware of the back-rotation of basis vectors and forward of regulars. Also, my plot results all came out reasonable so I don't think I could have made such a major error!