Initial Circular Orbit Research Articles

Exploratory scenarios for the differential nuclear and tidal evolution of TZ Fornacis

Aims. TZ Fornacis is a double-lined eclipsing binary system with similar masses (2.057 ± 0.001 and 1.958 ± 0.001 M⊙) but characterized by very different radii (8.28 ± 0.22 and 3.94 ± 0.17 R⊙). This similarity in terms of mass makes it possible to study the system’s differential stellar evolution as well as some aspects of its tidal evolution. With regard to its orbital elements, it was recently confirmed that its orbit is circular with an orbital period of 75.7 days. The less massive component rotates about 17 times faster than the primary one, which is synchronized with the mean orbital angular velocity. Our main objective in this work is to study both the nuclear and the tidal evolution of the system. Methods. To model the TZ For system, we used the MESA package, computing the grids using the exact observed masses, radii, and effective temperatures as input, and then varying the metallicity, the core overshooting amount, and the mixing-length parameter. A χ2 statistic was used to infer the optimal values of the core overshooting and the mixing-length parameters. The same procedure was used to generate rotating models with the GRANADA code. The respective errors in the average age of TZ For were less than 5%. On the other hand, the differential equations that govern the tidal evolution were integrated using the fifth-order Runge–Kutta method, ith a tolerance of 1 × 10−7. Results. We explored two scenarios regarding the initial eccentricities: a high one (0.30) and a case of an initial circular orbit. A good agreement has been found between the observational values of the eccentricity, synchronism levels, and orbital period with the values predicted by the integration of the tidal evolution equations. The influence of the friction timescale on the evolution of the orbital elements of TZ For is also studied here. The orbital elements most affected by the uncertainties in the friction timescale are the synchronism levels of the two components. On the other hand, we used the properties of the rotating models generated by the GRANADA code as the initial angular velocities instead of using trial values. In this case, comparisons between the theoretical values of the orbital elements and their observed counterparts also lead to a good interagreement.

Chaotic exits from a weakly magnetized Schwarzschild black hole

A charged particle kicked from an initial circular orbit around a weakly magnetized Schwarzschild black hole undergoes transient chaotic motion before either getting captured by the black hole or escaping upstream or downstream with respect to the direction of the magnetic field. These final states form basins of attraction in the space of initial states. We provide a detailed numerical study of the basin structure of this initial state space. We find it to possess the peculiar Wada property: each of its basin boundaries is shared by all three basins. Using basin entropy as a measure, we show that uncertainty in predicting the final exit state increases with stronger magnetic interaction. We also present an approximate analytic expression of the critical escape energy for a vertically-kicked charged particle, and discuss how this depends on the strength of the magnetic interaction.

Orbital plane change maneuver strategy using electric propulsion

The orbital dynamics basis for orbital plane change maneuver using chemical propulsion is the impulsive orbital change theory in practical engineering. The orbital plane change theory using continuous thrust suitable for electric propulsion is studied in this paper. A set of nonsingular orbital variational equations using quaternion is used to investigate the orbital motion of spacecraft under constant normal acceleration departing from a Keplerian circular orbit firstly. Results show that the orbit of spacecraft under continuous constant normal acceleration is a circular orbit over the gravitational center on the same spherical surface as the initial orbit, and it is tangent to the initial circular orbit at the initial position. Continuous and discontinuous maneuver strategies are then designed using the previous theory, and they are validated by numerical simulations. Results indicate that the strategy can work effectively.

Energy-Conserving Planar Spacecraft Motion with Constant-Thrust Acceleration

Orbital motion of spacecraft under the action of constant-thrust acceleration in the direction perpendicular to the velocity is studied. A spacecraft starting in an initial circular orbit obtains a minimum (perigee) radius along an inbound trajectory. If the thrust is maintained, the spacecraft returns to the initial circular orbit in an outbound trajectory. A maximum (apogee) radius is obtained when starting in a circular orbit along an outbound trajectory, after which the vehicle returns to its initial orbit. Only one integral of motion is known to exist in this system; therefore, a full analytical solution cannot be found. However, an analytical solution to the flight-direction angle (that is, the angle between the position and velocity vectors) is found in terms of the position magnitude. This solution leads to several insights, such as an analytical expression for the range of acceleration magnitudes that enables prograde motion. Periodic orbits are also found in this system for various values of acceleration, and their stability is studied. Even though a full analytical solution is not provided, a finite Taylor expansion about the acceleration magnitude yields an approximate analytical solution for the trajectory beginning on a circular orbit.

Tangent-impulse transfer from elliptic orbit to an excess velocity vector

The two-body orbital transfer problem from an elliptic parking orbit to an excess velocity vector with the tangent impulse is studied. The direction of the impulse is constrained to be aligned with the velocity vector, then speed changes are enough to nullify the relative velocity. First, if one tangent impulse is used, the transfer orbit is obtained by solving a single-variable function about the true anomaly of the initial orbit. For the initial circular orbit, the closed-form solution is derived. For the initial elliptic orbit, the discontinuous point is solved, then the initial true anomaly is obtained by a numerical iterative approach; moreover, an alternative method is proposed to avoid the singularity. There is only one solution for one-tangent-impulse escape trajectory. Then, based on the one-tangent-impulse solution, the minimum-energy multi-tangent-impulse escape trajectory is obtained by a numerical optimization algorithm, e.g., the genetic method. Finally, several examples are provided to validate the proposed method. The numerical results show that the minimum-energy multi-tangent-impulse escape trajectory is the same as the one-tangent-impulse trajectory.

Conditions of the Maximum Principle in the Problem of Optimal Control over an Aggregate of Dynamic Systems and Their Application to Solution of the Problems of Optimal Control of Spacecraft Motion

The necessary first-order conditions of strong local optimality (conditions of maximum principle) are considered for the problems of optimal control over a set of dynamic systems. To derive them a method is suggested based on the Lagrange principle of removing constraints in the problems on a conditional extremum in a functional space. An algorithm of conversion from the problem of optimal control of an aggregate of dynamic systems to a multipoint boundary value problem is suggested for a set of systems of ordinary differential equations with the complete set of conditions necessary for its solution. An example of application of the methods and algorithm proposed is considered: the solution of the problem of constructing the trajectories of a spacecraft flight at a constant altitude above a preset area (or above a preset point) of a planet's surface in a vacuum (for a planet with atmosphere beyond the atmosphere). The spacecraft is launched from a certain circular orbit of a planet's satellite. This orbit is to be determined (optimized). Then the satellite is injected to the desired trajectory segment (or desired point) of a flyby above the planet's surface at a specified altitude. After the flyby the satellite is returned to the initial circular orbit. A method is proposed of correct accounting for constraints imposed on overload (mixed restrictions of inequality type) and on the distance from the planet center: extended (nonpointlike) intermediate (phase) restrictions of the equality type.

Three-dimensional LEO keeping using low-thrust

The purpose of this paper is to investigate the two methods for low Earth orbit keeping: periodic low-thrust (PLT) method and continuous low-thrust (CLT) method. The perturbation forces which can cause deviation of the low Earth orbit and are considered include the centrifugal force and Coriolis force due to the Earth's rotation, the odd and even harmonics of the gravitational field, and the aerodynamic drag. The allowable value in the altitude drop is −5 km. In the PLT method, the thrust is turned on periodically when the altitude drop reaches the allowable value. Then it is turned off when the deviation is corrected to zero and another cycle of orbit keeping starts. In the CLT method, the thrust vector is set to compensate the vector sum of the perturbation forces so that the satellite remains in its initial circular orbit all the time. It is found that if the altitude of the initial circular orbit is 128 km, the two methods have the same fuel consumption. Below 128 km the CLT method must be used, and above 128 km the PLT method is much more economic. The same numerical computation has been performed when the allowable value in altitude drop is −10 km. We find the same characteristics. The change from CLT method to PLT method should occur at 135 km.