The file BMAN20_1 contains the periodic and Poisson terms of the rigid Mars nutation series BMAN20.1 the file BMAN20rs_1_RISE contains the periodic terms of the rigid Mars nutation series BMAN20RS.1.RISE the file BMAN20rs_1_VIKING contains the periodic terms of the rigid Mars nutation series BMAN20RS.1.VIKING the file BMAN20_1_PM_Ext contains the series for the Polar motion induced by the external gravitational torque Computational procedure explained in (Baland et al. 2020 and Yseboodt et al. 2023) BMAN20.1 is a follow-up of BMAN20 that includes the following corrections and modifications: - The four long-period terms of the original series are removed because their effect is included in the precession terms. - The series is now scaled to the precession rate of Mars by Konopliv et al. (2016), but expressed with respect to the mean orbit of J2000 instead of the mean orbit of 1980. Except for the geodetic contributions, the solution is multiplied by 0.9999096 with respect to BMAN20. - the two and four terms with period close to the ter and semi-annual periods, respectively, are now merged with the main ter and semi-annual terms. The difference with a solution where those terms are not merged is of 0.05 mas and 0.02 mas in longitude and obliquity, respectively, +/-50y from J2000. - Terms with the same periods but due to different physical causes (indirect and direct effects of Jupiter, Venus, and of the Earth) are now merged. Therefore, they do not pass the truncation criterion in the same way as in BMAN20. - Quadratic terms and Poisson terms in alpha and delta and prograde and retrograde terms, are corrected for second-order effects that were wrongly overlooked in BMAN20 (see Yseboodt et al., 2023) - The obliquity epoch value now includes a correction of -1.4 mas, corresponding to the constant term of the nutations series used in Konopliv et al. (2016). - Three terms due to the triaxiality of Mars are now included, instead of one. All triaxal terms are quasi semi-diurnal. Individually, the two new terms are just at the level of the truncation threshold, but their combination is well above the threshold. With a truncation criterion of 0.025 milliarcseconds in prograde and/or retrograde amplitude, we now identify 31 nutation terms. BMAN20RS.1.RISE is a modified version of the model, which reproduces at best the behavior of the BMAN20 solution around the 16th of August 2020, the mean epoch of the RISE mission. BMAN20RS.1.VIKING is a modified version of the model, which reproduces at best the behavior of the BMAN20 solution around the 10th of October 1977, the mean epoch of the VIKING mission. Like BMAN20RS, BMAN20RS.1.RISE and BMAN20RS.1.VIKING group the main periodic terms and their associated Poisson terms. Unlike BMAN20RS, BMAN20RS.1.RISE and BMAN20RS.1.VIKING contain all the small terms of the parent series, to ensure accuracy of the order of about 0.1 mas in the time domain et the mission epoch. The user is free to consider the number of terms he/she finds relevant for his/her applications. The series for the Polar motion induced by the external gravitational torque is a byproduct of the nutation theory. They are given for both the Rigid and Non Rigid cases. %------------------------------------------------------------------------------------------------------------------------- Truncation criterion: prograde and/or retrograde amplitude larger than 0.025 mas. Solution computed with HD=0.00537968, so that the total longitude variation rate is -7607.6118 mas/year (Konopliv et al. 2016, expressed with respect to the mean orbit of J2000). The solution can be rescaled to any value of HD. %------------------------------------------------------------------------------------------------------------------------- Columns signification (Nutation and Poisson terms): - j is the item number. - Ma, Ju, Sa, Te, Ve are the mean longitudes of Mars, Jupiter, Saturn, the Earth and of Venus. - phi is the rotation angle of Mars. - NPh and NDe are the mean longitudes of the nodes of Phobos and Deimos, respectively. - Periods are given in Earth solar days. - psiC and psiS (mas) are the cosine and sine amplitudes of the nutation in longitude of the angular momentum axis. - epsC and epsS (mas) are the cosine and sine amplitudes of the nutation in obliquity of the angular momentum axis. - P and R (mas) are the prograde and retrograde amplitudes of the nutation of the angular momentum axis. - pi and rho (deg) are the phases to be added to the phase of the prograde and retrograde terms, respectively. - alphaC and alphaS (mas) are the cosine and sine amplitudes of the nutation in right ascension of the angular momentum axis. - deltaC and deltaS (mas) are the cosine and sine amplitudes of the nutation in declination of the angular momentum axis. Columns signification (Polar motion terms): - Periods are given in Earth solar days. - X cos and X sin (mas) are the cosine and sine amplitudes of the X component of Polar Motion. - Y cos and Y so, (mas) are the cosine and sine amplitudes of the Y component of Polar Motion. %------------------------------------------------------------------------------------------------------------------------- The rotation axis can be assimilated to the angular momentum axis. The figure axis CANNOT be assimilated to the angular momentum axis. %------------------------------------------------------------------------------------------------------------------------- The values of the arguments are (Moisson and Bretagnon 2000: VSOP2000) (Jacobson and Lainey 2014: Phobos and Deimos ephemerides) (rotation rate of Mars by Konopliv et al. (2016) expressed with respect to the mean orbit of J2000) Sa = 0.87401678345 + 213.2990797783 T Ju = 0.59954667809 + 529.6909721118 T Ma = 6.20349959869 + 3340.6124347175 T Te = 1.75346994632 + 6283.0758504457 T Ve = 3.17613445715 + 10213.2855473855 T NPh= 2.13055663363 - 2779.4193805084 T NDe= 0.20283841509 - 114.7466716724 T phi= 3.63666363337 + 2236871.5239122 T where T is the dynamical time measured in thousand of years since J2000.0 %------------------------------------------------------------------------------------------------------------------------- For BMAN20.1, the waves in the time domain are obtained the following way: Delta psi = T^alpha ( psiC Cos(varphi) + psiS Sin(varphi) ) Delta eps = T^alpha ( epsC Cos(varphi) + epsS Sin(varphi) ) Delta alpha = T^alpha ( alphaC Cos(varphi) + alphaS Sin(varphi) ) Delta delta = T^alpha ( deltaC Cos(phvarphii) + deltaS Sin(varphi) ) with varphi = f t + varphi0, a linear combination of the arguments Sa, Ju, Ma, Te, Ve, NPh, NDe, and phi. f is positive. delta x = T^alpha ( P Cos(f t + pi) + R Cos (-f t - rho) ) delta y = T^alpha ( P Sin(f t + pi) + R Sin (-f t - rho) ) alpha in T^alpha is an integer in-between 0 and 1 (not the right ascension) For BMAN20.1, the first and second order Gamma coefficients used to transform the nutation and Poisson series between the eps/psi and alpha/delta formulations are provided. The formulas are given in Yseboodt et al. (2023). %------------------------------------------------------------------------------------------------------------------------- For BMAN20RS.1.RISE and BMAN20RS.1.VIKING, the waves in the time domain are obtained the following way: Delta psi = ( psiC Cos(varphi) + psiS Sin(varphi) ) Delta eps = ( epsC Cos(varphi) + epsS Sin(varphi) ) Delta alpha = ( alphaC Cos(varphi) + alphaS Sin(varphi) ) Delta delta = ( deltaC Cos(varphi) + deltaS Sin(varphi) ) An alternative representation is also provided, for which Delta psi = ( psiC Cos(f t) + psiS Sin(f t) ) Delta eps = ( epsC Cos(f t) + epsS Sin(f t) ) Delta alpha = ( alphaC Cos(f t) + alphaS Sin(f t) ) Delta delta = ( deltaC Cos(f t) + deltaS Sin(f t) ) The amplitudes in obliquity/longitude and in alpha/delta for the RS solutions are obtained as the sum of the nutation amplitudes and of the Poisson amplitudes multiplied by the mission reference time. For BMAN20RS.1.RISE and BMAN20RS.1.VIKING, the first order Gamma coefficients used to transform the nutation series between the eps/psi and alpha/delta formulations are provided. The first order Gamma coefficients for the RS solutions are defined in such a way that second order Gamma coefficients are not needed. %------------------------------------------------------------------------------------------------------------------------- For BMAN20_1_PM_Ext, the waves in the time domain are obtained the following way (alternative representation): X = ( Xcos Cos(f t) + Xsin Sin(f t) ) Y = ( Ycos Cos(f t) + Ysin Sin(f t) ) with f=2*pi/period %------------------------------------------------------------------------------------------------------------------------- The J2000 epoch values, expressed with respect to the mean orbit of J2000, are (in degrees) psi0 = 35.49752578 eps0 = 25.19181935 alpha0 = 317.68111503 delta0 = 52.88635277 %------------------------------------------------------------------------------------------------------------------------- The secular and quadratic terms of the BMAN20.1 solution are (in mas): (-7.613585EO6 T - 14352.4 T^2) in longitude for the Solar torque (-2 T ) in longitude for the Solar torque, long period terms (+6754 T) in longitude (geodetic precession) (-235 T) in longitude for Phobos torque (-201 T) in longitude for Deimos torque (-342 T) in longitude for Planet torque (-7.607612E06 T - 14352.4 T^2) Total in longitude Note that the total precession in longitude is the observed rate of -7607.6118 mas/year (Konopliv et al. 2016, expressed with respect to the mean orbit of J2000) (-2 T + 2007.3 T^2) in obliquity for the Solar torque (-1 T ) in obliquity for the Solar torque, long period terms ( 0 T) in longitude (geodetic precession) ( 0 T) in longitude for Phobos torque ( 0 T) in longitude for Deimos torque (-3 T) in obliquity for Planet torque (-6 T + 2007.3 T^2) Total in obliquity Note that the total precession in obliquity is NOT the observed rate of -2.078 mas/year (Konopliv et al. 2016, expressed with respect to the mean orbit of J2000). Such a large obliquity rate is not expected from the modelization. If the total in obliquity is the one of the model, then (-3.909057E06 T - 10884.9 T^2) Total in right ascension (-2.218619E06 T + 15889.7 T^2) Total in declination If the total in obliquity is (-2078 T + 2007.3 T^2), with the observed rate, then (-3.911410E06 T - 10805.8 T^2) Total in right ascension (-2.217109EO6 T + 15915.6 T^2) Total in declination The epoch values, secular and quadratic terms of the BMAN20.1 solution can be used without additional modification for any RS solution, since the quadratic terms can be considered to be known with a sufficient accuracy from the rigid theory, which is not the case for the periodic terms which also depends on (unknown) liquid core amplification. %------------------------------------------------------------------------------------------------------------------------- Any comments or request could be send to Baland Rose-Marie at the following e-mail address: Rose-Marie.Baland@oma.be Last revision: 17/04/23.