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Appendix a on the Rotation Axis of The App endix A On The Rotation Axis of the Sun In interpreting measurements of solar velo cities, it is crucial to know the direction of the Sun's axis of rotation for each observation. The \known" axis of solar rotation that is used almost universally to daywas determined almost 150 years ago Carrington, 1863 by observing the motions of sunsp ots recorded on photographic plates at a 1 ground-based observatory . Although Carrington did not o er an estimate of the uncertainty in his prop osed standard, it seems unlikely that the error was less than ab out a tenth of a degree. Subsequent researchers have attempted to improve on the accuracy of this measurement, with varying results. This app endix presents a new measurement whichmay o er the most accurate determination yet, and compares the results with those obtained by other researchers in the last one hundred years. Finally, it is prop osed that the set of more mo dern measurements b e used to adopt a new standard for the Sun's axis of rotation. 1 It is interesting to note that the observations required to make this calculation were available at least twohundred years earlier see Eddy et al. 1976 119 APPENDIX A. ON THE ROTATION AXIS OF THE SUN 120 A.1 The Co ordinate System: i and 2 Consider the representation of the Sun in gure A.1 . The great circle EN is the intersection of the plane of the ecliptic the plane of the Earth's orbit with the solar surface. The p oint C de nes the center of the sphere, and CK is p erp endicular to the ecliptic. The p oint is de ned such that a straight line drawn from C to p oints in the direction of the vernal equinox; this is the reference direction from which celestial longitudes are measured. The line joining C and the Earth cuts the sphere at E , and the plane de ned by CK and CT , p erp endicular to CE, is de ned as the plane of the disk. The helio centric ecliptic longitude of the Earth is E; the geo centric longitude of the Sun is denoted by = E + 180 . The Sun rotates around an axis CP , where P is the north p ole of the Sun. The great circle p erp endicular to this axis is the solar equator, of which the arc NJ is a section. The inclination of the solar equator to the ecliptic the angle JNE in gure A.1 is denoted by i. The p oint N is the ascending no de of the solar equator on the ecliptic plane, and its longitude N is given the symb ol . The values of i and completely determine the Sun's rotation axis. The standard values used by the astronomical community are those determined by Carrington 1863, namely i = 7:25 ; c A.1 = 73:67 +0:013958 t 1850:0; c where t is expressed in years. The value of increases slowly due to precession. The p oint E will b e seen as the center of the disk; the latitude of this p ointmust b e known in order to correctly interpret the measured velo cities. This latitude is denoted by B .For the spherical triangle de ned by PEN, 0 2 The notation and description in this section b orrow heavily from Smart 1977. It is included here so that the reader need not make reference to that work. APPENDIX A. ON THE ROTATION AXIS OF THE SUN 121 K P C T ϒ E F N J Figure A.1: The geometry of the co ordinate system used in measuring solar p ositions. The line CP represents the rotation axis of the Sun; the axis CK is the p ole of the ecliptic. The inclination angle KCP is exaggerated. Other symb ols are explained in the text. APPENDIX A. ON THE ROTATION AXIS OF THE SUN 122 K P′ y C Figure A.2: The angle denoted by y is the e ective position angle of the solar image 0 taken by MDI. The p oint P is the pro jection of the north p ole of the Sun into the plane of the image. PN = 90 ; PE = 90 B ; 0 A.2 EN = 180 ; 6 PNE = 90 i: Using the cosine law for spherical triangles see e.g. Smart 1977, the angle B can 0 thus b e computed from the rotation elements: sin B = sin sin i A.3 0 APPENDIX A. ON THE ROTATION AXIS OF THE SUN 123 Now consider the angle made by the pro jection of the rotation axis CP onto the 3 plane of the disk, and the p ole of the ecliptic CK. Let us denote this angle by y see gure A.2. The axes CE, CT , and CK de ne a rectangular co ordinate system, and the co ordinates of P are given by sin KP cos EF; sin KP sin EF; cos KP . Hence the pro jection of P onto the CT -CK plane has co ordinates sin KP sin EF; cos KP , and the angle y is given by tan y = tan KP sin EF . The angle KP is the inclination i, and EF can b e expressed in terms of such that tan y = cos tan i A.4 Equations A.3 and A.4 thus de ne the co ordinate system on the disk as seen from Earth. A.2 SOHO Orbit and Alignment The SOHO spacecraft orbits the Sun in a halo orbit ab out the Earth-Sun Lagrange p oint L . The spacecraft attitude is maintained via star trackers so that the MDI 1 camera is aligned always with the Carrington rotation elements. More precisely, if the Sun's rotation axis is accurately describ ed by the elements i ; of equation A.1, c c then SOHO is aligned such that the angle y describ ed by equation A.4 will always b e zero. On the other hand, what if the Sun rotates ab out an axis describ ed by co ordinates i; which are slightly di erent from the Carrington elements? Clearly this will mean that the p osition angle y and the heliographic latitude of disk center B will b e 0 somewhat di erent from their exp ected values. To quantify these e ects, supp ose that the actual rotation elements of the Sun di er from the Carrington elements according to 3 Note that y is not the p osition angle P of the rotation axis as it is usually de ned. APPENDIX A. ON THE ROTATION AXIS OF THE SUN 124 i = i +i; c A.5 = + : c De ne the angle errors E y y ; P c A.6 E B B : B 0 0c Here y is given by equation A.4, with i replaced by i and replaced by . The c c c latitude B is de ned in the same wayby equation A.3. 0c Using Taylor series expansions for the trigonometric functions, the equations A.3 and A.4 can b e used to express y and B in terms of the known values y and B 0 c 0c and the errors i and . The de nitions A.6 can then b e used to write 2 tan i sin +i sec i cos c c c c A.7 E ' P 2 2 1 + tan i cos c c i cos i sin sin i cos c c c c E ' A.8 B 1=2 2 2 [1 sin i sin ] c c Here all terms which are second-order or higher in i, , E , and E have b een P B neglected, since all of these quantities can b e assumed to b e quite small. Furthermore, the quantity i is small enough that terms which are second-order in i can also b e c c ignored; in this case equations A.7 and A.8 simplify to E ' tan i sin +i cos A.9 P c c c E ' i cos i sin sin i cos A.10 B c c c c Hence it can b e seen that the angle errors E and E will vary with a one-year p erio d B P as SOHO orbits around the Sun. This fact provides the basis for the measurement of i and outlined in the following section. It is worth noting that some other sources of these angle errors | for example, a xed error in the alignment of MDI relative to the SOHO axis | do not show a similar dep endence on orbital p osition. APPENDIX A. ON THE ROTATION AXIS OF THE SUN 125 A.3 Solar Velo cities and Angle Errors An error in the p osition angle or in the heliographic latitude of disk center will result in errors in the analysis of solar velo cities. For example, if the p osition angle has a small error E , then the \meridional" velo city determined in the analysis will P contain a comp onent of the zonal velo city or rotation which leaks in: v =v +v sin E A.11 ;m P Here the subscript m denotes a measured velo city, whereas values without such nota- tion are the true velo cities in the and directions. For the rotation, the principal source of error will b e a misidenti cation of the latitude of each p osition on the disk: 0 v =v v E A.12 ;m B 0 where v is the derivative of the zonal velo city with resp ect to colatitude .
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