A new perturbative solution to the motion around triangular Lagrangian points in the elliptic restricted three-body problem

The equations of motion of the planar elliptic restricted three-body problem are transformed to four decoupled Hill’s equations. By using the Floquet theorem, a perturbative solution to the oscillator equations with time-dependent periodic coefficients are presented. We clarify the transformation details that provide the applicability of the method. The form of newly derived equations inherently comprises the stability boundaries around the triangular Lagrangian points. The analytic approach is valid for system parameters $$0 < e \le 0.05$$ 0 < e ≤ 0.05 and $$0 < \mu \le 0.01$$ 0 < μ ≤ 0.01 where e denotes the eccentricity of the primaries, while $$\mu $$ μ is the mass parameter. Possible application to known extrasolar planetary systems is also demonstrated.

Possible Advantages of a Twin Spacecraft Heliospheric Mission at the Sun-Earth Lagrangian Points L4 and L5

After the launch of STEREO twin spacecraft, and most recently of Solar Orbiter and Parker Solar Probe spacecraft, the next mission that will explore Sun-Earth interactions and how the Sun modulates the Heliosphere will be the “Lagrange” mission, which will consist of two satellites placed in orbit around L1 and L5 Sun-Earth Lagrangian points. Despite the significant novelties that will be provided by such a double vantage point, there will be also missing information, that are briefly discussed here. For future heliospheric missions, an alternative advantageous approach that has not been considered so far would be to place two twin spacecraft not in L1 and L5, but in L4 and L5 Lagrangian points. If these two spacecraft will be equipped with in situ instruments, and also remote sensing instruments measuring not only photospheric but also coronal magnetic fields, significant advancing will be possible. In particular, data provided by such a twin mission will allow to follow the evolution of magnetic fields from inside the Sun (with stereoscopic helioseismology), to its surface (with classical photospheric magnetometers), and its atmosphere (with spectro-polarimeters); this will provide a tremendous improvement in our physical understanding of solar activity. Moreover, the L4-L5 twin satellites will take different interesting configurations, such as relative quadrature, and quasi-quadrature with the Earth, providing a baseline for monitoring the Sun-to-Earth propagation of solar disturbances.

Unveiling Perturbing Effects of P-R Drag on Motion around Triangular Lagrangian Points of the Photogravitational Restricted Problem of Three Oblate Bodies

The perturbing effects of the Poynting-Robertson drag on motion of an infinitesimal mass around triangular Lagrangian points of the circular restricted three-body problem under small perturbations in the Coriolis and centrifugal forces when the three bodies are oblate spheroids and the primaries are emitters of radiation pressure, is the focus of this paper. The equations governing the dynamical system have been derived and locations of triangular Lagrangian points are determined. It is seen that the locations are influenced by the perturbing forces of centrifugal perturbation and the oblateness, radiation pressure and, P-R drag of the primaries. Using the software Mathematica, numerical analysis are carried out to demonstrate how the dynamical elements: mass ratio, oblateness, radiation pressure, P-R drag and centrifugal perturbation influence the positions of triangular equilibrium points, zero velocity surfaces and the stability. Our investigation reveals that, though the radiation pressure, oblateness and centrifugal perturbation decrease region of stability when motion is stable, however, they are not the influential forces of instability but the P-R drag. In the region when motion around the triangular points are stable an inclusion of the P-R drag of the bigger primary even by an almost negligible value of 1.04548*10-9 overrides other effect and changes stability to instability. Hence, we conclude that the P-R drag is a strong perturbing force which changes stability to instability and motion around triangular Lagrangian points remain unstable in the presence of the P-R drag.

Dust trapping around Lagrangian points in protoplanetary disks

Aims. Trojans are defined as objects that share the orbit of a planet at the stable Lagrangian points L4 and L5. In the Solar System, these bodies show a broad size distribution ranging from micrometer (μm) to centimeter (cm) particles (Trojan dust) and up to kilometer (km) rocks (Trojan asteroids). It has also been theorized that earth-like Trojans may be formed in extra-solar systems. The Trojan formation mechanism is still under debate, especially theories involving the effects of dissipative forces from a viscous gaseous environment. Methods. We perform hydro-simulations to follow the evolution of a protoplanetary disk with an embedded 1–10 Jupiter-mass planet. On top of the gaseous disk, we set a distribution of μm–cm dust particles interacting with the gas. This allows us to follow dust dynamics as solids get trapped around the Lagrangian points of the planet. Results. We show that large vortices generated at the Lagrangian points are responsible for dust accumulation, where the leading Lagrangian point L4 traps a larger amount of submillimeter (submm) particles than the trailing L5, which traps mostly mm–cm particles. However, the total bulk mass, with typical values of ~Mmoon, is more significant in L5 than in L4, in contrast to what is observed in the current Solar System a few gigayears later. Furthermore, the migration of the planet does not seem to affect the reported asymmetry between L4 and L5. Conclusions. The main initial mass reservoir for Trojan dust lies in the same co-orbital path of the planet, while dust migrating from the outer region (due to drag) contributes very little to its final mass, imposing strong mass constraints for the in situ formation scenario of Trojan planets.

Location and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid

The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit for stability remains the same.

Locations of Lagrangian points and periodic orbits around triangular points in the photo gravitational elliptic restricted three-body problem with oblateness

Locations of the Lagrangian points are computed and periodic orbits are studied around the triangular points in the photogravitational elliptic restricted three-body problem (ER3BP) by considering the more massive primary as the source of radiation and smaller primary as an oblate spheroid. A new mean motion taken from Sharma et al. [13] is used to study the effect of radiation pressure and oblateness of the primaries. The critical mass parameter that bifurcates periodic orbits from non-periodic orbits tends to reduce with radiation pressure and oblateness. The transition curves defining stable region of orbits are drawn for different values of radiation pressure and oblateness using the analytical method of Bennet [14]. Tadpole orbits with long- and short- periodic oscillations are obtained for Sun-Jupiter and Sun-Saturn systems.