The current LAGEOS–LARES 2 experiment aims to accurately measure the general relativistic Lense–Thirring effect in the gravitomagnetic field of the spinning Earth generated by the latter’s angular momentum $${\varvec{J}}$$ . The key quantity to a priori analytically assess the overall systematic uncertainty is the ratio $${\mathcal {R}}^{J_2}$$ of the sum of the classical precessions of the satellites’ nodes $$\Omega $$ induced by the Earth’s oblateness $$J_2$$ to the sum of their post-Newtonian counterparts. , if the sum of the inclinations of both satellites were $$180^\circ $$ , the semimajor axes and the eccentricities being , $${\mathcal {R}}^{J_2}$$ would vanish. Actually, it is so by a large amount because of the departures of the satellites’ orbital configurations from the ones. Thus, $$J_2$$ impacts not only directly through its own uncertainty, but also through the errors in all the other physical and orbital parameters entering $${\mathcal {R}}^{J_2}$$ . The consequences of this fact are examined in greater details than done so far in the literature. The Van Patten and Everitt’s proposal in 1976 of looking at the sum of the node precessions of two counter-orbiting spacecraft in (low-altitude) circular polar orbits is revamped rebranding it POLAr RElativity Satellites (POLARES). Regardless the specific type of satellite and tracking technologies that may be eventually adopted, it be superior to the LAGEOS–LARES 2 one from the point of view of the orbital characteristics since, given the semimajor axes and eccentricities of the existing laser-ranged cousins, its $${\mathcal {R}}^{J_2}$$ is sensitive to the impact of the deviations from its orbital configuration.

Italy

Lorenzo Iorio

Will LAGEOS and LARES 2 succeed in accurately measuring frame-dragging?

To the first post-Newtonian order, if two test particles revolve in opposite directions about a massive, spinning body along two circular and equatorial orbits with the same radius, they take different times to return to the reference direction relative to which their motion is measured: it is the so-called gravitomagnetic clock effect. The satellite moving in the same sense of the rotation of the primary is slower, and experiences a retardation with respect to the case when the latter does not spin, while the one circling in the opposite sense of the rotation of the source is faster, and its orbital period is shorter than it would be in the static case. The resulting time difference due to the stationary gravitomagnetic field of the central spinning body is proportional to the angular momentum per unit mass of the latter through a numerical factor which so far has been found to be  $$4\pi $$  <math> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </math>  . A numerical integration of the equations of motion of a fictitious test particle moving along a circular path lying in the equatorial plane of a hypothetical rotating object by including the gravitomagnetic acceleration to the first post-Newtonian order shows that, actually, the gravitomagnetic corrections to the orbital periods are larger by a factor of 4 in both the prograde and retrograde cases. Such an outcome, which makes the proportionality coefficient of the gravitomagnetic difference in the orbital periods of the two counter-revolving orbiters equal to  $$16\pi $$  <math> <mrow> <mn>16</mn> <mi>π</mi> </mrow> </math>  , confirms an analytical calculation recently published in the literature by the present author. It is an important result in view of the astrophysical implications of the gravitomagnetic clock effect around Kerr black holes.

Revisiting the gravitomagnetic clock effect

Erratum to: Is it possible to measure new general relativistic third-body effects on the orbit of Mercury with BepiColombo?

We numerically and analytically work out the first-order post-Newtonian (1pN) orbital effects induced on the semimajor axis a, the eccentricity e, the inclination I, the longitude of the ascending node $$\Omega $$ <math><mi>Ω</mi></math> , the longitude of perihelion $$\varpi $$ <math><mi>ϖ</mi></math> , and the mean longitude at epoch $$\epsilon $$ <math><mi>ϵ</mi></math>  of a test particle orbiting its primary, assumed static and spherically symmetric, by a distant massive third body X. For Mercury, the rates of change of the linear trends found are $$\dot{I}_\mathrm {1pN}^\mathrm {X} = -4.3\,\mathrm {microarcseconds\,per\,century}\,\left( \upmu \mathrm {as\,cty}^{-1}\right) $$ <math><mrow><msubsup><mover><mi>I</mi><mo>˙</mo></mover><mrow><mn>1</mn><mi>pN</mi></mrow><mi>X</mi></msubsup><mo>=</mo><mo>-</mo><mn>4.3</mn><mspace width="0.166667em"></mspace><mrow><mi>microarcseconds</mi><mspace width="0.166667em"></mspace><mi>per</mi><mspace width="0.166667em"></mspace><mi>century</mi></mrow><mspace width="0.166667em"></mspace><mfenced><mi>μ</mi><msup><mrow><mi>as</mi><mspace width="0.166667em"></mspace><mi>cty</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></mrow></math> , $${\dot{\Omega }}_\mathrm {1pN}^\mathrm {X} = 18.2\,\upmu \mathrm {as\,cty}^{-1}$$ <math><mrow><msubsup><mover><mi>Ω</mi><mo>˙</mo></mover><mrow><mn>1</mn><mi>pN</mi></mrow><mi>X</mi></msubsup><mo>=</mo><mn>18.2</mn><mspace width="0.166667em"></mspace><mi>μ</mi><msup><mrow><mi>as</mi><mspace width="0.166667em"></mspace><mi>cty</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></math> , $${\dot{\varpi }}_\mathrm {1pN}^\mathrm {X} = 30.4\,\upmu \mathrm {as\,cty}^{-1}$$ <math><mrow><msubsup><mover><mi>ϖ</mi><mo>˙</mo></mover><mrow><mn>1</mn><mi>pN</mi></mrow><mi>X</mi></msubsup><mo>=</mo><mn>30.4</mn><mspace width="0.166667em"></mspace><mi>μ</mi><msup><mrow><mi>as</mi><mspace width="0.166667em"></mspace><mi>cty</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></math> , $${\dot{\epsilon }}_\mathrm {1pN}^\mathrm {X} = 271.4\,\upmu \mathrm {as\,cty}^{-1}$$ <math><mrow><msubsup><mover><mi>ϵ</mi><mo>˙</mo></mover><mrow><mn>1</mn><mi>pN</mi></mrow><mi>X</mi></msubsup><mo>=</mo><mn>271.4</mn><mspace width="0.166667em"></mspace><mi>μ</mi><msup><mrow><mi>as</mi><mspace width="0.166667em"></mspace><mi>cty</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></math> , respectively. Such values, which are due to the added actions of the other planets from Venus to Saturn, are essentially at the same level of, or larger by one order of magnitude than, the latest formal errors in the Hermean orbital precessions calculated with the EPM2017 ephemerides. The perihelion precession $${\dot{\varpi }}_\mathrm {1pN}^\mathrm {X}$$ <math><msubsup><mover><mi>ϖ</mi><mo>˙</mo></mover><mrow><mn>1</mn><mi>pN</mi></mrow><mi>X</mi></msubsup></math>  turns out to be smaller than some values recently appeared in the literature in view of a possible measurement with the ongoing BepiColombo mission. Linear combinations of the supplementary advances of the Keplerian orbital elements for several planets, if determined experimentally by the astronomers, could be set up in order to disentangle the 1pN N-body effects of interest from the competing larger precessions like those due to the Sun’s quadrupole moment $$J_2$$ <math><msub><mi>J</mi><mn>2</mn></msub></math>  and angular momentum $$\varvec{S}$$ <math><mrow><mi>S</mi></mrow></math> .

New general relativistic contributions to Mercury’s orbital elements and their measurability

The distinction between the mean anomaly $${\mathcal {M}}(t)$$ <math><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></math>  and the mean anomaly at epoch $$\eta $$ <math><mi>η</mi></math> , and the mean longitude l(t) and the mean longitude at epoch $$\epsilon $$ <math><mi>ϵ</mi></math>  is clarified in the context of a their possible use in post-Keplerian tests of gravity, both Newtonian and post-Newtonian. In particular, the perturbations induced on $${\mathcal {M}}(t),\,\eta ,\,l(t),\,\epsilon $$ <math><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace width="0.166667em"></mspace><mi>η</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>l</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mspace width="0.166667em"></mspace><mi>ϵ</mi></mrow></math>  by the post-Newtonian Schwarzschild and Lense–Thirring fields, and the classical accelerations due to the atmospheric drag and the oblateness $$J_2$$ <math><msub><mi>J</mi><mn>2</mn></msub></math>  of the central body are calculated for an arbitrary orbital configuration of the test particle and a generic orientation of the primary’s spin axis $$\varvec{{\hat{S}}}$$ <math><mrow><mover><mi>S</mi><mo>^</mo></mover></mrow></math> . They provide us with further observables which could be fruitfully used, e.g., in better characterizing astrophysical binary systems and in more accurate satellite-based tests around major bodies of the Solar System. Some erroneous claims by Ciufolini and Pavlis appeared in the literature are confuted. In particular, it is shown that there are no net perturbations of the Lense–Thirring acceleration on either the semimajor axis a and the mean motion $$n_{\mathrm{b}}$$ <math><msub><mi>n</mi><mi>b</mi></msub></math> . Furthermore, the quadratic signatures on $${\mathcal {M}}(t)$$ <math><mrow><mi>M</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></math>  and l(t) due to certain disturbing non-gravitational accelerations like the atmospheric drag can be effectively disentangled from the post-Newtonian linear trends of interest provided that a sufficiently long temporal interval for the data analysis is assumed. A possible use of $$\eta $$ <math><mi>η</mi></math>  along with the longitudes of the ascending node $$\Omega $$ <math><mi>Ω</mi></math>  in tests of general relativity with the existing LAGEOS and LAGEOS II satellites is suggested.

On the mean anomaly and the mean longitude in tests of post-Newtonian gravity

We consider a binary system composed of a pulsar and a massive, fast rotating, highly distorted main sequence star of mass M, spin angular momentum $$\mathbf S $$ <math><mi>S</mi></math> , dimensionless mass quadrupole moment $$J_2$$ <math><msub><mi>J</mi><mn>2</mn></msub></math> , equatorial and polar radii $$R_\text {e},~R_\text {p}$$ <math><mrow><msub><mi>R</mi><mtext>e</mtext></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>R</mi><mtext>p</mtext></msub></mrow></math> , flattening $$\nu \doteq (R_\text {e}-R_\text {p})/R_\text {e}$$ <math><mrow><mi>ν</mi><mo>≐</mo><mrow><mo>(</mo><msub><mi>R</mi><mtext>e</mtext></msub><mo>-</mo><msub><mi>R</mi><mtext>p</mtext></msub><mo>)</mo></mrow><mo>/</mo><msub><mi>R</mi><mtext>e</mtext></msub></mrow></math> , and ellipticity $$\varepsilon \doteq \sqrt{1-R_\text {p}^2/R_\text {e}^2}$$ <math><mrow><mi>ε</mi><mo>≐</mo><msqrt><mrow><mn>1</mn><mo>-</mo><msubsup><mi>R</mi><mtext>p</mtext><mn>2</mn></msubsup><mo>/</mo><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup></mrow></msqrt></mrow></math>  as a potential scenario to dynamically put to the test certain post-Keplerian effects of both Newtonian and post-Newtonian nature. We numerically produce time series of the perturbations $$\Delta \left( \delta \tau \right) $$ <math><mrow><mi>Δ</mi><mfenced><mi>δ</mi><mi>τ</mi></mfenced></mrow></math>  of the Rømer-like, orbital component of the pulsar’s time delay $$\delta \tau $$ <math><mrow><mi>δ</mi><mi>τ</mi></mrow></math>  induced over 10 years by the pN gravitoelectric mass monopole $$\left( \text {Schwarzschild}, GMc^{-2}\right) $$ <math><mfenced><mtext>Schwarzschild</mtext><mo>,</mo><mi>G</mi><mi>M</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced></math> , quadrupole $$\left( GMR^2_\text {e}J_2 c^{-2}\right) $$ <math><mfenced><mi>G</mi><mi>M</mi><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup><msub><mi>J</mi><mn>2</mn></msub><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced></math> , gravitomagnetic spin dipole $$\left( \text {Lense}{-}\text {Thirring},~GSc^{-2}\right) $$ <math><mfenced><mtext>Lense</mtext><mo>-</mo><mtext>Thirring</mtext><mo>,</mo><mspace width="3.33333pt"></mspace><mi>G</mi><mi>S</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced></math>  and octupole $$\left( GSR^2_\text {e}\varepsilon ^2 c^{-2}\right) $$ <math><mfenced><mi>G</mi><mi>S</mi><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup><msup><mi>ε</mi><mn>2</mn></msup><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced></math>  accelerations along with the Newtonian quadrupolar $$\left( GMR^2_\text {e}J_2\right) $$ <math><mfenced><mi>G</mi><mi>M</mi><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup><msub><mi>J</mi><mn>2</mn></msub></mfenced></math>  one. We do not deal with the various propagation time delays due to the travelling electromagnetic waves. It turns out that, for a Be-type star with $$M = 15~\text {M}_\odot ,~R_\text {e} = 5.96~\text {R}_\odot ,~\nu = 0.203,~S = 3.41\times $$ <math><mrow><mi>M</mi><mo>=</mo><mn>15</mn><mspace width="3.33333pt"></mspace><msub><mtext>M</mtext><mo>⊙</mo></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>R</mi><mtext>e</mtext></msub><mo>=</mo><mn>5.96</mn><mspace width="3.33333pt"></mspace><msub><mtext>R</mtext><mo>⊙</mo></msub><mo>,</mo><mspace width="3.33333pt"></mspace><mi>ν</mi><mo>=</mo><mn>0.203</mn><mo>,</mo><mspace width="3.33333pt"></mspace><mi>S</mi><mo>=</mo><mn>3.41</mn><mo>×</mo></mrow></math>  $$10^{45}~\text {J}~\text {s},\,J_2 = 1.92\times 10^{-3}$$ <math><mrow><msup><mn>10</mn><mn>45</mn></msup><mspace width="3.33333pt"></mspace><mtext>J</mtext><mspace width="3.33333pt"></mspace><mtext>s</mtext><mo>,</mo><mspace width="0.166667em"></mspace><msub><mi>J</mi><mn>2</mn></msub><mo>=</mo><mn>1.92</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow></math>  orbited by a pulsar with an orbital period $$P_\mathrm{b}\simeq $$ <math><mrow><msub><mi>P</mi><mi>b</mi></msub><mo>≃</mo></mrow></math>  40–70 days, the classical oblateness-driven effects are at the $$\lesssim 4-150~\text {s}$$ <math><mrow><mo>≲</mo><mn>4</mn><mo>-</mo><mn>150</mn><mspace width="3.33333pt"></mspace><mtext>s</mtext></mrow></math>  level, while the pN shifts are of the order of $$ \lesssim 1.5-20~\text {s}~\left( GMc^{-2}\right) ,~ \lesssim 10-40~\text {ms}~\left( GMR^2_\text {e} J_2 c^{-2}\right) ,$$ <math><mrow><mo>≲</mo><mn>1.5</mn><mo>-</mo><mn>20</mn><mspace width="3.33333pt"></mspace><mtext>s</mtext><mspace width="3.33333pt"></mspace><mfenced><mi>G</mi><mi>M</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mo>,</mo><mspace width="3.33333pt"></mspace><mo>≲</mo><mn>10</mn><mo>-</mo><mn>40</mn><mspace width="3.33333pt"></mspace><mtext>ms</mtext><mspace width="3.33333pt"></mspace><mfenced><mi>G</mi><mi>M</mi><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup><msub><mi>J</mi><mn>2</mn></msub><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mo>,</mo></mrow></math>  $$~ \lesssim 0.5-6~\text {ms}~\left( GSc^{-2}\right) ,~ \lesssim 5-20~\upmu \text {s}~\left( GSR^2_\text {e}\varepsilon ^2 c^{-2}\right) $$ <math><mrow><mspace width="3.33333pt"></mspace><mo>≲</mo><mn>0.5</mn><mo>-</mo><mn>6</mn><mspace width="3.33333pt"></mspace><mtext>ms</mtext><mspace width="3.33333pt"></mspace><mfenced><mi>G</mi><mi>S</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced><mo>,</mo><mspace width="3.33333pt"></mspace><mo>≲</mo><mn>5</mn><mo>-</mo><mn>20</mn><mspace width="3.33333pt"></mspace><mi>μ</mi><mtext>s</mtext><mspace width="3.33333pt"></mspace><mfenced><mi>G</mi><mi>S</mi><msubsup><mi>R</mi><mtext>e</mtext><mn>2</mn></msubsup><msup><mi>ε</mi><mn>2</mn></msup><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mfenced></mrow></math> , depending on their orbital configuration. The root-mean-square (rms) timing residuals $$\upsigma _{\tau }$$ <math><msub><mi>σ</mi><mi>τ</mi></msub></math>  of almost all the existing non-recycled, non-millisecond pulsars orbiting massive, fast rotating main sequence stars are $$\lesssim \text {ms}$$ <math><mrow><mo>≲</mo><mtext>ms</mtext></mrow></math> . Thus, such kind of binaries have the potential to become interesting laboratories to measure, or, at least, constrain, some Newtonian and post-Newtonian ($$GMc^{-2},\,GMJ_2c^{-2}$$ <math><mrow><mi>G</mi><mi>M</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>,</mo><mspace width="0.166667em"></mspace><mi>G</mi><mi>M</mi><msub><mi>J</mi><mn>2</mn></msub><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></math> , and, perhaps, $$GSc^{-2}$$ <math><mrow><mi>G</mi><mi>S</mi><msup><mi>c</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></math>  as well) key features of the distorted gravitational fields of the fast rotating stars hosted by them.

France

Michel Rieutord

Jean-Pierre Rozelot

Armando Souza

Classical and general relativistic post-Keplerian effects in binary pulsars hosting fast rotating main sequence stars

The inclination I of an Earth’s satellite in polar orbit undergoes a secular De Sitter precession of $$-\,7.6$$ <math><mrow><mo>-</mo><mspace width="0.166667em"></mspace><mn>7.6</mn></mrow></math>  milliarcseconds per year for a suitable choice of the initial value of its non-circulating node $$\Omega $$ <math><mi>Ω</mi></math> . The competing long-periodic harmonic rates of change of I due to the even and odd zonal harmonics of the geopotential vanish for either a circular or polar orbit, while no secular rates occur at all. This may open up, in principle, the possibility of measuring the geodesic precession in the weak-field limit with an accurately tracked satellite by improving the current bound of $$9\times 10^{-4}$$ <math><mrow><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup></mrow></math>  from Lunar Laser Ranging, which, on the other hand, may be even rather optimistic, by one order of magnitude, or, perhaps, even better. The most insidious competing effects are due to the solid and ocean components of the $$K_1$$ <math><msub><mi>K</mi><mn>1</mn></msub></math>  tide since their perturbations have nominal huge amplitudes and the same temporal pattern of the De Sitter signature. They vanish for polar orbits. Departures of $$\simeq {10^{-5}}^\circ \mathrm{to} {10^{-3}}^\circ $$ <math><mrow><mo>≃</mo><msup><mrow><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup></mrow><mo>∘</mo></msup><mi>to</mi><msup><mrow><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></mrow><mo>∘</mo></msup></mrow></math>  from the ideal polar geometry allow to keep the $$K_1$$ <math><msub><mi>K</mi><mn>1</mn></msub></math>  tidal perturbations to a sufficiently small level. Most of the other gravitational and non-gravitational perturbations vanish for the proposed orbital configuration, while the non-vanishing ones either have different temporal signatures with respect to the De Sitter effect or can be modeled with sufficient accuracy. In order to meet the proposed goal, the measurement accuracy of I should be better than $$\simeq 35~\text {microarcseconds}=0.034~\text {milliarcseconds}$$ <math><mrow><mo>≃</mo><mn>35</mn><mspace width="3.33333pt"></mspace><mtext>microarcseconds</mtext><mo>=</mo><mn>0.034</mn><mspace width="3.33333pt"></mspace><mtext>milliarcseconds</mtext></mrow></math>  over, say, 5 year.

Measuring the De Sitter precession with a new Earth’s satellite to the $$\simeq 10^{-5}$$ <math><mrow><mo>≃</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup></mrow></math>  level: a proposal

Recently, Will calculated an additional contribution to the Mercury’s precession of the longitude of perihelion $$\varpi $$ <math><mi>ϖ</mi></math>  of the order of $${\dot{\varpi }}_\text {W}\simeq 0.22~\text {milliarcseconds per century}$$ <math><mrow><msub><mover><mi>ϖ</mi><mo>˙</mo></mover><mtext>W</mtext></msub><mo>≃</mo><mn>0.22</mn><mspace width="3.33333pt"></mspace><mtext>milliarcseconds per century</mtext></mrow></math>  ($$\text {mas cty}^{-1}$$ <math><mrow><mtext>mas</mtext><mspace width="0.333333em"></mspace><msup><mtext>cty</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></math> ). It is partly a direct consequence of certain 1pN third-body accelerations entering the planetary equations of motion, and partly an indirect, mixed effect due to the simultaneous interplay of the standard 1pN pointlike acceleration of the primary with the Newtonian N-body acceleration, to the quadrupole order, in the analytical calculation of the secular perihelion precession with the Gauss equations. We critically discuss the actual measurability of the mixed effects with respect to direct ones. The current uncertainties in either the magnitude of the Sun’s angular momentum $$S_\odot $$ <math><msub><mi>S</mi><mo>⊙</mo></msub></math>  and the orientation of its spin axis $${\varvec{\hat{S}}}_\odot $$ <math><msub><mrow><mover><mi>S</mi><mo>^</mo></mover></mrow><mo>⊙</mo></msub></math>  impact the precessions $${\dot{\varpi }}_{J_2^\odot },~{\dot{\varpi }}_\text {LT}$$ <math><mrow><msub><mover><mi>ϖ</mi><mo>˙</mo></mover><msubsup><mi>J</mi><mn>2</mn><mo>⊙</mo></msubsup></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mover><mi>ϖ</mi><mo>˙</mo></mover><mtext>LT</mtext></msub></mrow></math>  induced by the Sun’s quadrupole mass moment and angular momentum via the Lense–Thirring effect to a level which makes almost impossible to measure $${\dot{\varpi }}_\text {W}$$ <math><msub><mover><mi>ϖ</mi><mo>˙</mo></mover><mtext>W</mtext></msub></math> , even in the hypothesis that it comes entirely from the aforementioned 1pN third-body accelerations. On the other hand, from the point of view of the Lense–Thirring effect itself, the mismodeled quadrupolar precession $$\delta {\dot{\varpi }}_{J_2^\odot }$$ <math><mrow><mi>δ</mi><msub><mover><mi>ϖ</mi><mo>˙</mo></mover><msubsup><mi>J</mi><mn>2</mn><mo>⊙</mo></msubsup></msub></mrow></math>  due to the uncertainties in $${\varvec{\hat{S}}}_\odot $$ <math><msub><mrow><mover><mi>S</mi><mo>^</mo></mover></mrow><mo>⊙</mo></msub></math>  corresponds to a bias of $$\simeq 9\%$$ <math><mrow><mo>≃</mo><mn>9</mn><mo>%</mo></mrow></math>  of the relativistic one. The resulting simulated mismodeled range and range-rate times series of BepiColombo are at about the per cent level of the nominal gravitomagnetic ones.

Is it possible to measure new general relativistic third-body effects on the orbit of Mercury with BepiColombo?

We develop a general approach to analytically calculate the perturbations ÎÎ´Ïp of the orbital component of the change Î´Ïp of the times of arrival of the pulses emitted by a binary pulsar p induced by the post-Keplerian accelerations due to the mass quadrupole Q2 , and the post-Newtonian gravitoelectric (GE) and LenseâThirring (LT) fields. We apply our results to the so-far still hypothetical scenario involving a pulsar orbiting the supermassive black hole in the galactic center at Sgr A â . We also evaluate the gravitomagnetic and quadrupolar Shapiro-like propagation delays Î´Ïprop . By assuming the orbit of the existing main sequence star S2 and a time span as long as its orbital period Pb , we obtain ÎÎ´ÏpGEâ²103s,ÎÎ´ÏpLTâ²0.6s,ÎÎ´ÏpQ2â²0.04s . Faster Pb=5years and more eccentric e=0.97 orbits would imply net shifts per revolution as large as ÎÎ´ÏpGEâ²10Ms,ÎÎ´ÏpLTâ²400s,ÎÎ´ÏpQ2â²103s , depending on the other orbital parameters and the initial epoch. For the propagation delays, we have Î´ÏpropLTâ²0.02s,Î´ÏpropQ2â²1Î¼s . The results for the mass quadrupole and the LenseâThirring field depend, among other things, on the spatial orientation of the spin axis of the Black Hole. The expected precision in pulsar timing in Sgr A â is of the order of 100Î¼s , or, perhaps, even 1â10Â Î¼s . Our method is, in principle, neither limited just to some particular orbital configuration nor to the dynamical effects considered in the present study.

Post-Keplerian perturbations of the orbital time shift in binary pulsars: an analytical formulation with applications to the galactic center

In the framework of the emergent gravity scenario by Verlinde, it was recently observed by Liu and Prokopec that, among other things, an anomalous pericenter precession would affect the orbital motion of a test particle orbiting an isolated central body. Here, it is shown that, if it were real, its expected magnitude for the inner planets of the Solar System would be at the same level of the present-day accuracy in constraining any possible deviations from their standard perihelion precessions as inferred from long data records spanning about the last century. The most favorable situation for testing the Verlinde-type precession seems to occur for Mars. Indeed, according to recent versions of the EPM and INPOP planetary ephemerides, non-standard perihelion precessions, of whatsoever physical origin, which are larger than some ≈ 0.02–0.11 milliarcseconds per century are not admissible, while the putative precession predicted by Liu and Prokopec amounts to 0.09 milliarcseconds per century. Other potentially interesting astronomical and astrophysical scenarios like, e.g., the Earth’s LAGEOS II artificial satellite, the double pulsar system PSR J0737-3039A/B and the S-stars orbiting the Supermassive Black Hole in Sgr A ∗ are, instead, not viable because of the excessive smallness of the predicted precessions for them.