## Orbital Mechanics and Astrodynamics Techniques and Tools for Space Missions Gerald R. Hintz

To determine the arrival date at the target body. To provide a preliminary estimate of the amount of propellant to be carried onboard the spacecraft. To provide information on the capture orbit when the spacecraft arrives at the target body. Springer International Publishing Switzerland G. Download sample.

Then, once the accelerations are given, it is necessary to use integral calculus in order to get from the second derivatives to the positions. In a more general context, where the mass may be changing with time, such as happens with an extended application of thrust to a vehicle, with the gradual reduction of weight as fuel is used up, or in cases of relativistic speeds, the force is given by the first derivative of momentum, but the principle is the same.

In the case of the 2-body problem, where the only force involved is the gravitational attraction between the two bodies, it is frequently said that Newton was able to give a complete solution. That is not, strictly speaking, the case, if one means by "a solution" of a differential equation, an expression for the unknown function whose derivatives appear in the equation.

### Reward Yourself

In this case, it would mean finding an expression for the position as a function of time. However, what Newton showed was that the orbit of each of the bodies lies on a conic section in a fixed inertial frame of reference , and in the case considered by Kepler, where the orbit as an ellipse, there is an explicit expression for the time as a function of the position. What one wants, of course, is x as a function of t, and much effort and ingenuity has gone into finding effective means of solving Kepler's equation for x in terms of t.

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Lagrange did extensive work on the problem, in the course of which he developed both Fourier series and Bessel functions, named after later mathematicians who investigated these concepts in greater detail. Both Laplace and Gauss made major contributions, and succeeding generations continued to work on the subject. When there are more than two bodies involved, the problem cannot be solved analytically; instead, the integration positions from accelerations must be done numerically: now, with high-speed computers.

So, numerical integral calculus is a major factor of spacecraft navigation. One may picture navigation as being the modeling of mother nature on acomputer.

At some time, with the planets in their orbits, a spacecraft is given a push outward into the solar system. Its subsequent orbit is then determined by the gravitational forces upon it due to the sun and planets.

## orbital mechanics

We compute these, step-by-step in time, seeing how the changing forces determine the motion of the spacecraft. This is very similar to what one may picture being done in nature. How does one get an accurate orbit in the computer? The spacecraft's orbit is measured as it progresses on its journey, and the computer model is adjusted in order to best fit the actual measurements.

Here one uses another type of calculus: estimation theory. It involves changing the initial "input parameters" starting positions and velocities into the computer in order to make the "output parameters" positions and velocities at subsequent times match what is being measured: adjusting the computer model to better fit reality. Also in navigation, one must "reduce" the measurements.

Usually, the measurements don't correspond exactly with the positions in the computer; one must apply a few formulae before a comparison can be made. For instance, the positions in the computer represent the centers of mass of the different planets; a radar echo, however, measures the path from the radio antenna to the spot on a planet's surfaces from which the signal bounces back to earth.

This processing involves the use of trigonometry, geometry, and physics. Finally, there is error analysis, or "covariance" calculus. In the initial planning stages of a mission, one is more interested in how accurately we will know the positions of the spacecraft and its target, not in the exact positions themselves. With low accuracy, greater amounts of fuel are required, and it could be that some precise navigation would not even be possible.

## Flight Mechanics Laboratory | NASA

Covariance analysis takes into account 1 what measurements we will have of the spacecraft: how many and how good, 2 how accurately we will be able to compute the forces, and 3 how accurately we will know the position of the target. These criteria are then used in order to determine how closely we can deliver the spacecraft to the target. Again, poor accuracy will require more fuel to correct the trajectory once the spacecraft starts approaching its final target.

One of the mathematical tools used to optimize some feature of a flight trajectory, such as fuel consumption or flight time, is a maximum principle introduced by Pontryagin in Despite its conceptual simplicity, huge engineering challenges have to be overcome.

Our research combines innovative technology, modern orbital dynamics and systems engineering in a multi-disciplinary optimisation approach, in order to select the most advantageous design points of future solar power satellites. This research theme covers a number of topics related to the theoretical and numerical study of spacecraft orbital dynamics. We are particularly interested in spacecraft which have a large cross-sectional area with respect to their mass. One family is made of the so-called solar sails, in which solar radiation pressure is collected by a large lightweight reflective membrane deployed from the spacecraft.

Solar sailing is a very promising technology for spacecraft propulsion.

These devices provide a small, but continuous, acceleration to the spacecraft. As a consequence, the resulting orbits do not follow well-known Keplerian laws, and the same happens when considering multi-body environments like the Sun-Earth or Sun-Moon systems. For these reasons, we are interested in techniques for space mission design and trajectory optimisation. We design attitude control and estimation algorithms. In the past, we developed detumbling and sun tracking algorithms for UKube Our main research interest is implementing efficient attitude control and estimation algorithms for small satellites.

We are also investigating the control of swarms and constellations of spacecraft.