REFERENCE FUNCTIONAL AND CHARACTERISTIC SPACE FOR LAGRANGE AND BERNSTEIN OPERATORS

This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.

A method for calculating probability of collision between space objects

A method is developed to calculate probability of collision. Based on geometric features of space objects during the encounter, it is reasonable to separate the radial orbital motions from those in the cross section for most encounter events that occur in a near-circular orbit. Therefore, the probability of collision caused by differences in both altitude of the orbit in the radial direction and the probability of collision caused by differences in arrival time in the cross section are calculated. The net probability of collision is expressed as an explicit expression by multiplying the above two components. Numerical cases are applied to test this method by comparing the results with the general method. The results indicate that this method is valid for most encounter events that occur in near-circular orbits.

Relativistic transformation between τ and TCG for Mars missions under IAU Resolutions

Considering the fact that the general theory of relativity has become an in- extricable part of deep space missions, we investigate the relativistic transformation between the proper time of an onboard clock τ and the Geocentric Coordinate Time （TCG） for Mars missions. By connecting τ with this local timescale associated with the Earth, we extend previous works which focus on the transformation between τ and the Barycentric Coordinate Time （TCB）. （TCB is the global coordinate time for the whole solar system.） For practical convenience, the relation between τ and TCG is recast to directly depend on quantities which can be read from ephemerides. We find that the difference between τ and TCG can reach the level of about 0.2 seconds in a year. To distinguish various sources in the transformation, we numerically calculate the contributions caused by the Sun, eight planets, three large asteroids and the space- craft. It is found that if the threshold of 1 microsecond is adopted, this transformation must include effects due to the Sun, Venus, the Moon, Mars, Jupiter, Saturn and the velocities of the spacecraft and Earth.

Relativistic transformation between τ and TCB for Mars missions: Fourier analysis on its accessibility with clock offset

In the context of the fact that Einstein＇s general relativity has become an inevitable part of deep space missions, we will extend previous works on relativistic transformation between the proper time ^- of a clock onboard a spacecraft orbiting Mars and the Barycentric Coordinate Time （TCB） by taking the clock offset into ac- count and investigate its accessibility by Fourier analysis on the residuals after fitting the ^--TCB curve in terms of n-th order polynomials. We find that if the accuracy of a clock can achieve better than ~ 10-5 s or ~ 10-6 s （depending on the type of clock offset） in one year after calibration, the relativistic effects on the difference between 7- and TCB will need to be carefully considered.

Relativistic algorithm for time transfer in Mars missions under IAU Resolutions: an analytic approach

With tremendous advances in modem techniques, Einstein＇s general rela- tivity has become an inevitable part of deep space missions. We investigate the rela- tivistic algorithm for time transfer between the proper time - of the onboard clock and the Geocentric Coordinate Time, which extends some previous works by including the effects of propagation of electromagnetic signals. In order to evaluate the implicit algebraic equations and integrals in the model, we take an analytic approach to work out their approximate values. This analytic model might be used in an onboard com- puter because of its limited capability to perform calculations. Taking an orbiter like Yinghuo-1 as an example, we find that the contributions of the Sun, the ground station and the spacecraft dominate the outcomes of the relativistic corrections to the model.

Relativistic time transfer for a Mars lander:from proper time to Areocentric Coordinate Time

As the first step in relativistic time transfer for a Mars lander from its proper time to the time scale at the ground station, we investigate the transformation between proper time and Areocentric Coordinate Time （TCA） in the framework of IAU Resolutions. TCA is a local time scale for Mars, which is analogous to the Geocentric Coordinate Time （TCG） for Earth. This transformation contains two contributions： inter- hal and external. The internal contribution comes from the gravitational potential and the rotation of Mars. The external contribution is due to the gravitational fields of other bodies （except Mars） in the Solar System. When the （in）stability of an onboard clock is assumed to be at the level of 10-13, we find that the internal contribution is dominated by the gravitational potential of spherical Mars with necessary corrections asso- ciated with the height of the lander on the areoid, the dynamic form factor of Mars, the flattening of the areoid and the spin rate of Mars. For the external contribution, we find the gravitational effects from other bodies in the Solar System can be safely neglected in this case after calculating their maximum values.

Relativistic time transfer for a Mars lander： from Areocentric CoordinateTime to Barycentric Coordinate Time

As the second step of relativistic time transfer for a Mars lander,we investigate the transformation between Areocentric Coordinate Time（TCA）and Barycentric Coordinate Time（TCB）in the framework of IAU Resolutions.TCA is a local time scale for Mars,which is analogous to the Geocentric Coordinate Time（TCG）for Earth.This transformation has two parts:contributions associated with gravitational bodies and those depending on the position of the lander.After setting the instability of an onboard clock to 10;and considering that the uncertainty in time is about 3.2 microseconds after one Earth year,we find that the contributions of the Sun,Mars,Jupiter and Saturn in the leading term associated with these bodies can reach a level exceeding the threshold and must be taken into account.Other terms can be safely ignored in this transformation for a Mars lander.