The Solar Lense-Thirring Effect and BepiColombo

The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The Solar Lense-Thirring Effect and BepiColombo Based on: L. Iorio, Mon. Not. Roy. Astron. Soc., 476(2), 1811-1825, 2018 The analytical approach BepiColombo Conclusions References Lorenzo Iorio Ministero dell’Istruzione, dell’Università e della Ricerca (M.I.U.R.)-Istruzione, Editor-in-Chief of Fifteenth Marcel Grossmann Meeting - MG15 University of Rome "La Sapienza" - Rome, July 1-7, 2018 Outline The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate 1 Two-body range and range-rate The analytical approach BepiColombo 2 The analytical approach Conclusions References 3 BepiColombo 4 Conclusions 5 References The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Two-body range and range-rate • Let us consider a pair of test particles A, B orbiting the same spinning, oblate primary characterized by quadrupole moment J2 and angular momentum S. • The two-body range ρ and range-rate ρ̇ contain a wealth of information about the physical properties of the primary’s gravitational field. • We analytically calculate the perturbations ∆ρ, ∆ρ̇ induced by the primary’s J2 and Lense-Thirring field, proportional to S, for an arbitrary orientation of its spin axis Ŝ and without making a-priori simplifying assumptions on the orbital configurations of A, B. We apply our results to the forthcoming BepiColombo mission to Mercury in view of a possible test of the Lense-Thirring effect which, on the other hand, acts as a bias if one is, instead, interested in the Sun’s J2 [Iorio 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Two-body range and range-rate • Let us consider a pair of test particles A, B orbiting the same spinning, oblate primary characterized by quadrupole moment J2 and angular momentum S. • The two-body range ρ and range-rate ρ̇ contain a wealth of information about the physical properties of the primary’s gravitational field. • We analytically calculate the perturbations ∆ρ, ∆ρ̇ induced by the primary’s J2 and Lense-Thirring field, proportional to S, for an arbitrary orientation of its spin axis Ŝ and without making a-priori simplifying assumptions on the orbital configurations of A, B. We apply our results to the forthcoming BepiColombo mission to Mercury in view of a possible test of the Lense-Thirring effect which, on the other hand, acts as a bias if one is, instead, interested in the Sun’s J2 [Iorio 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Two-body range and range-rate • Let us consider a pair of test particles A, B orbiting the same spinning, oblate primary characterized by quadrupole moment J2 and angular momentum S. • The two-body range ρ and range-rate ρ̇ contain a wealth of information about the physical properties of the primary’s gravitational field. • We analytically calculate the perturbations ∆ρ, ∆ρ̇ induced by the primary’s J2 and Lense-Thirring field, proportional to S, for an arbitrary orientation of its spin axis Ŝ and without making a-priori simplifying assumptions on the orbital configurations of A, B. We apply our results to the forthcoming BepiColombo mission to Mercury in view of a possible test of the Lense-Thirring effect which, on the other hand, acts as a bias if one is, instead, interested in the Sun’s J2 [Iorio 2018]. The range perturbation ∆ρ The Solar Lense-Thirring effect and BepiColombo L. Iorio The two-body range perturbation ∆ρ is [Cheng 2002] is Two-body range and range-rate The analytical approach ∆ρ = (∆rA − ∆rB ) · ρ̂, where BepiColombo Conclusions References (1) ρ̂ = (rA − rB ) 2 , ρ = (rA − rB ) · (rA − rB ) . ρ (2) In the RTN scheme, the change ∆r of the position r is ∆r = ∆R R̂ + ∆T T̂ + ∆N N̂, (3) General expressions for ∆R, ∆T , ∆N, in terms of the shifts ∆κ(f ), κ = a, e, I, Ω, ω, M, are in [Casotto 1993]. They are calculated for given accelerations A (Lense-Thirring, J2 , etc.) by integrating the Gauss equations for κ̇ from f0 to f onto a Keplerian ellipse as reference orbit. The range-rate perturbation ∆ρ̇ The Solar Lense-Thirring effect and BepiColombo L. Iorio The two-body range-rate perturbation ∆ρ̇ is [Cheng 2002] is Two-body range and range-rate The analytical approach ∆ρ̇ = (∆vA − ∆vB ) · ρ̂ + (∆rA − ∆rB ) · ρ̂v , where BepiColombo Conclusions References (4) ρ̂v = (vA − vB ) − ρ̇ρ̂ , ρ̇ = (vA − vB ) · ρ̂. ρ (5) In the RTN scheme, the change ∆v of the velocity v is ∆v = ∆vR R̂ + ∆vT T̂ + ∆vN N̂, (6) General expressions for ∆vR , ∆vT , ∆vN , in terms of the shifts ∆κ(f ), κ = a, e, I, Ω, ω, M, are in [Casotto 1993]. They are calculated for given accelerations A (Lense-Thirring, J2 , etc.) by integrating the Gauss equations for κ̇ from f0 to f onto a Keplerian ellipse as reference orbit. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References The opportunity offered by BepiColombo • BepiColombo, to be launched in 2018, should greatly improve the ephemerides of Mercury to the σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 level. • Let us apply our analytical results of Sec. 2 to the Earth-Mercury range and range-rate during the planned extended phase (2026 March 14-2028 May 1). • In Figs 1 to 4, we plot the Lense-Thirring and J2⊙ signatures and for different values of its uncertainty σJ ⊙ . The nominal J2⊙ signals are much larger than the 2 nominal Lense-Thirring ones since their amplitudes can reach 300 m and 3 × 10−2 cm s−1 , while the relativistic ones amount to 10 m and 10−3 cm s−1 . But, by assuming σJ ⊙ = 5.5 × 10−10 [Imperi et al. 2018], its 2 signatures reduce to 0.6 m and 10−4 cm s−1 (Fig. 4). Fig. 5 shows that our analytical time series agree well with the numerically integrated ones. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References The opportunity offered by BepiColombo • BepiColombo, to be launched in 2018, should greatly improve the ephemerides of Mercury to the σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 level. • Let us apply our analytical results of Sec. 2 to the Earth-Mercury range and range-rate during the planned extended phase (2026 March 14-2028 May 1). • In Figs 1 to 4, we plot the Lense-Thirring and J2⊙ signatures and for different values of its uncertainty σJ ⊙ . The nominal J2⊙ signals are much larger than the 2 nominal Lense-Thirring ones since their amplitudes can reach 300 m and 3 × 10−2 cm s−1 , while the relativistic ones amount to 10 m and 10−3 cm s−1 . But, by assuming σJ ⊙ = 5.5 × 10−10 [Imperi et al. 2018], its 2 signatures reduce to 0.6 m and 10−4 cm s−1 (Fig. 4). Fig. 5 shows that our analytical time series agree well with the numerically integrated ones. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References The opportunity offered by BepiColombo • BepiColombo, to be launched in 2018, should greatly improve the ephemerides of Mercury to the σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 level. • Let us apply our analytical results of Sec. 2 to the Earth-Mercury range and range-rate during the planned extended phase (2026 March 14-2028 May 1). • In Figs 1 to 4, we plot the Lense-Thirring and J2⊙ signatures and for different values of its uncertainty σJ ⊙ . The nominal J2⊙ signals are much larger than the 2 nominal Lense-Thirring ones since their amplitudes can reach 300 m and 3 × 10−2 cm s−1 , while the relativistic ones amount to 10 m and 10−3 cm s−1 . But, by assuming σJ ⊙ = 5.5 × 10−10 [Imperi et al. 2018], its 2 signatures reduce to 0.6 m and 10−4 cm s−1 (Fig. 4). Fig. 5 shows that our analytical time series agree well with the numerically integrated ones. Nominal LT and J2⊙ signals The Solar Lense-Thirring effect and BepiColombo L. Iorio Earth-Mercury range perturbations Earth-Mercury range-rate perturbations 300 0.03 BepiColombo 0.02  Δρ ( cm s-1 ) The analytical approach 200 100 Δρ (m) Two-body range and range-rate 0.01 0 -100 -200 0.00 LT J2⊙ = 2.295×10-7 LT -0.01 -300 Conclusions J2⊙ = 2.295×10-7 -0.02 2026.5 2027.0 2027.5 t (yr) 2028.0 2026.5 2027.0 2027.5 2028.0 t (yr) References Figure 1: Nominal Lense-Thirring and J2⊙ analytical shifts of the Earth-Mercury range (in m) and range-rate (in cm s−1 ) during the expected extended mission of BepiColombo from 2026 March 14 to 2028 May 1. For the Sun’s angular momentum and quadrupole moment, the nominal values S⊙ = 190.0 × 1039 kg m2 s−1 [Pijpers 1998] and J2⊙ = 2.295 × 10−7 [Viswanathan et al. 2017] were adopted. The nominal values of the right ascension and declination of the Sun’s spin axis which were used are [Seidelmann et al. 2007] α⊙ = 286.13 deg, δ⊙ = 63.87 deg. Nominal LT and mismodelled J2⊙ signals The Solar Lense-Thirring effect and BepiColombo L. Iorio Earth-Mercury range perturbations Earth-Mercury range-rate perturbations 10 0.0010  Δρ ( cm s-1 ) 5 Δρ (m) Two-body range and range-rate 0.0005 0 The analytical approach 0.0000 -5 LT BepiColombo Conclusions References -10 -0.0005 σJ ⊙ = 9×10-9 2 2026.5 -0.0010 2027.0 2027.5 t (yr) 2028.0 LT σJ ⊙ = 9×10-9 2 2026.5 2027.0 2027.5 2028.0 t (yr) Figure 2: Nominal Lense-Thirring and mismodelled J2⊙ analytical shifts of the Earth-Mercury range (in m) and range-rate (in cm s−1 ) during the expected extended mission of BepiColombo from 2026 March 14 to 2028 May 1. For the Sun’s quadrupole moment, the mismodelled value σJ ⊙ = 9 × 10−9 2 [Park et al. 2017] was adopted. The nominal values of the right ascension and declination of the Sun’s spin axis which were used are [Seidelmann et al. 2007] α⊙ = 286.13 deg, δ⊙ = 63.87 deg. Nominal LT and mismodelled J2⊙ signals The Solar Lense-Thirring effect and BepiColombo The analytical approach BepiColombo Conclusions Earth-Mercury range-rate perturbations 0.0005 5  Δρ ( cm s-1 ) Two-body range and range-rate Earth-Mercury range perturbations 10 Δρ (m) L. Iorio 0.0000 0 -0.0005 -5 LT -10 σJ ⊙ = 1×10-9 -0.0010 2 2026.5 2027.0 2027.5 t (yr) 2028.0 LT σJ ⊙ = 1×10-9 2 2026.5 2027.0 2027.5 2028.0 t (yr) References Figure 3: Nominal Lense-Thirring and mismodelled J2⊙ analytical shifts of the Earth-Mercury range (in m) and range-rate (in cm s−1 ) during the expected extended mission of BepiColombo from 2026 March 14 to 2028 May 1. For the Sun’s quadrupole moment, the mismodelled value σJ ⊙ = 1 × 10−9 2 [Viswanathan et al. 2017] was adopted. The nominal values of the right ascension and declination of the Sun’s spin axis which were used are [Seidelmann et al. 2007] α⊙ = 286.13 deg, δ⊙ = 63.87 deg. Nominal LT and mismodelled J2⊙ signals The Solar Lense-Thirring effect and BepiColombo L. Iorio Earth-Mercury range perturbations Earth-Mercury range-rate perturbations 10 0.0005  Δρ ( cm s-1 ) 5 Δρ (m) Two-body range and range-rate 0.0000 0 The analytical approach -0.0005 -5 LT BepiColombo Conclusions References -10 σJ ⊙ = 5.5×10-10 -0.0010 2 2026.5 2027.0 2027.5 t (yr) 2028.0 LT σJ ⊙ = 5.5×10-10 2 2026.5 2027.0 2027.5 2028.0 t (yr) Figure 4: Nominal Lense-Thirring and mismodelled J2⊙ analytical shifts of the Earth-Mercury range (in m) and range-rate (in cm s−1 ) during the expected extended mission of BepiColombo from 2026 March 14 to 2028 May 1. For the Sun’s quadrupole moment, the mismodelled value σJ ⊙ = 5.5 × 10−10 2 [Imperi et al. 2018] was adopted. The nominal values of the right ascension and declination of the Sun’s spin axis which were used are [Seidelmann et al. 2007] α⊙ = 286.13 deg, δ⊙ = 63.87 deg. LT numerical vs analytical time series The Solar Lense-Thirring effect and BepiColombo L. Iorio BepiColombo Conclusions Earth-Mercury LT range-rate perturbation   Δρ anal -Δρ num ( cm s-1 ) The analytical approach Δρ anal -Δρ num (m) Two-body range and range-rate Earth-Mercury LT range perturbation 0.00005 0 0.00010 0.00005 0.00000 -0.00005 -0.00010 -0.00005 2026.5 2027.0 2027.5 t (yr) 2028.0 -0.00015 2026.5 2027.0 2027.5 2028.0 t (yr) References Figure 5: Differences between the analytical and the numerical nominal Lense-Thirring shifts of the Earth-Mercury range (in m) and range-rate (in cm s−1 ) during the expected extended mission of BepiColombo from 2026 March 14 to 2028 May 1. For the Sun’s angular momentum, the nominal value S⊙ = 190.0 × 1039 kg m2 s−1 [Pijpers 1998] was adopted. The nominal values of the right ascension and declination of the Sun’s spin axis which were used are [Seidelmann et al. 2007] α⊙ = 286.13 deg, δ⊙ = 63.87 deg. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Summary and overview • We analytically calculated the range and range-rate shifts due to S and J2 of the primary for an arbitrary orientation of its symmetry axis Ŝ and a generic orbit. • We applied our results to the planned extended phase (2026-2028) of BepiColombo. Its expected range and range-rate accuracy is σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 . • The nominal Lense-Thirring signatures amount to ∆ρLT = 10 m, ∆ρ̇LT = 10−3 cm s−1 . • The J2⊙ signatures, if modelled to the σJ2 = 1 × 10−9 level [Viswanathan et al. 2017], would be 10 times smaller than the Lense-Thirring ones. It is expected to reach even the σJ ⊙ = 5.5 × 10−10 level 2 [Imperi et al. 2018]. • A major source of error, not treated here, is the uncertainty in the position of Ŝ [Schettino et al. 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Summary and overview • We analytically calculated the range and range-rate shifts due to S and J2 of the primary for an arbitrary orientation of its symmetry axis Ŝ and a generic orbit. • We applied our results to the planned extended phase (2026-2028) of BepiColombo. Its expected range and range-rate accuracy is σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 . • The nominal Lense-Thirring signatures amount to ∆ρLT = 10 m, ∆ρ̇LT = 10−3 cm s−1 . • The J2⊙ signatures, if modelled to the σJ2 = 1 × 10−9 level [Viswanathan et al. 2017], would be 10 times smaller than the Lense-Thirring ones. It is expected to reach even the σJ ⊙ = 5.5 × 10−10 level 2 [Imperi et al. 2018]. • A major source of error, not treated here, is the uncertainty in the position of Ŝ [Schettino et al. 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Summary and overview • We analytically calculated the range and range-rate shifts due to S and J2 of the primary for an arbitrary orientation of its symmetry axis Ŝ and a generic orbit. • We applied our results to the planned extended phase (2026-2028) of BepiColombo. Its expected range and range-rate accuracy is σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 . • The nominal Lense-Thirring signatures amount to ∆ρLT = 10 m, ∆ρ̇LT = 10−3 cm s−1 . • The J2⊙ signatures, if modelled to the σJ2 = 1 × 10−9 level [Viswanathan et al. 2017], would be 10 times smaller than the Lense-Thirring ones. It is expected to reach even the σJ ⊙ = 5.5 × 10−10 level 2 [Imperi et al. 2018]. • A major source of error, not treated here, is the uncertainty in the position of Ŝ [Schettino et al. 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Summary and overview • We analytically calculated the range and range-rate shifts due to S and J2 of the primary for an arbitrary orientation of its symmetry axis Ŝ and a generic orbit. • We applied our results to the planned extended phase (2026-2028) of BepiColombo. Its expected range and range-rate accuracy is σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 . • The nominal Lense-Thirring signatures amount to ∆ρLT = 10 m, ∆ρ̇LT = 10−3 cm s−1 . • The J2⊙ signatures, if modelled to the σJ2 = 1 × 10−9 level [Viswanathan et al. 2017], would be 10 times smaller than the Lense-Thirring ones. It is expected to reach even the σJ ⊙ = 5.5 × 10−10 level 2 [Imperi et al. 2018]. • A major source of error, not treated here, is the uncertainty in the position of Ŝ [Schettino et al. 2018]. The Solar Lense-Thirring effect and BepiColombo L. Iorio Two-body range and range-rate The analytical approach BepiColombo Conclusions References Summary and overview • We analytically calculated the range and range-rate shifts due to S and J2 of the primary for an arbitrary orientation of its symmetry axis Ŝ and a generic orbit. • We applied our results to the planned extended phase (2026-2028) of BepiColombo. Its expected range and range-rate accuracy is σρ ≃ 0.1 m, σρ̇ ≃ 10−4 cm s−1 . • The nominal Lense-Thirring signatures amount to ∆ρLT = 10 m, ∆ρ̇LT = 10−3 cm s−1 . • The J2⊙ signatures, if modelled to the σJ2 = 1 × 10−9 level [Viswanathan et al. 2017], would be 10 times smaller than the Lense-Thirring ones. It is expected to reach even the σJ ⊙ = 5.5 × 10−10 level 2 [Imperi et al. 2018]. • A major source of error, not treated here, is the uncertainty in the position of Ŝ [Schettino et al. 2018]. References The Solar Lense-Thirring effect and BepiColombo L. Iorio L. Iorio, Mon. Not. Roy. Astron. Soc., 476, 1811, 2018 Two-body range and range-rate M.K. Cheng, J. Geod., 76, 169, 2002 The analytical approach S. Casotto, Celest. Mech. Dyn. Astron., 55, 209, 1993 BepiColombo Conclusions References F.P. Pijpers, Mon. Not. Roy. Astron. Soc., 297, L76, 1998 V. Viswanathan, et al., Notes Sci. Tech. Inst Méc. Céleste, 108, 2017 P.K. Seidelmann, et al., Celest. Mech. Dyn. Astr., 98, 155, 2007 R.S. Park, et al., Astron. J., 153, 121, 2017 L. Imperi, et al., Icarus, 301, 9, 2018 G. Schettino, et al., arXiv:1804.02996, 2018