where e is the eccentricity of the orbit and J is the angular momentum of the orbit. This may be further written as e = —GM/2a where a = ^yf{e2 — 1) is the semi-major axis. This is an important result in classical mechanics: The energy of an elliptical orbit depends upon the semi-major axis. Hence for a planet its energy/mass is e = -v2/r — GM/r and the same energy per mass obtains for a spacecraft in this kissing orbit at its semi-major exist e = —GM/2a2. It it then apparent that the delta-vee required for the orbital transfer is

It is worth noting in equation 5.5 that for e = 1 the energy is zero, which geometrically corresponds to a parabolic orbit for the escape trajectory. For e = 0 the orbit is circular and the energy is maximal. For comets the eccentricity is very close to unity. It has been suggested that interstellar space is populated with small ice bodies which occasionally swing by a star to become a transient comet with e > 1.

The Hohmann transfer orbit is the minimal energy configuration [5.1]. It is largely employed for interplanetary space missions. It is also used in the reentry of manned spacecraft. A delta-vee on the shuttle will put it on an elliptical orbit that kisses its more circular orbit around the Earth. This brings it within the Earth's atmosphere which is what really breaks the velocity of the spacecraft. This also illustrates why it takes energy to send spacecraft to the inner solar system. The extreme case is a suggestion to send garbage to the sun. To do this a delta-vee of 29.5km/sec is required to put the payload in a dead stop relative to the heliocentric coordinates. From here the payload would then fall into the sun. This is a large change in velocity and at the extreme end of current launch vehicle capability.

For a transfer orbit to a planet on a different orbital plane than the Earth this requires a non-Hohmann orbital transfer. This requires an acceleration that changes the energy/mass by v • a/a2. This is an added complexity that will not be explored here. Obviously for more muscular thrusters the transfer is also non-Hohmann. The details of these aspects of astrodynamics are left to the reader to explore further.

Astrodynamics for star travel is comparatively simple. All one needs is to compute the average velocity of the spacecraft and the relative velocity of the target star. It is then a simple matter to figure the direction the craft should be aimed to intersect the star. Interstellar space flight involves high velocities in a region where gravity is not a significant factor.

The classical mechanics developed to this point permits us to mention why systems with three or more bodies can't be solved in closed form. For any N body problem there are 3 equations for the center of mass, 3 for the momentum, 3 for the angular momentum and one for the energy. These are 10 constraints on the problem. An N-body problem has 6N degrees of freedom. For N = 2 this means the solution is given by a first integral with degree 2. For a three body problem this first integral has degree 8. This runs into the problem that Galois illustrated which is that any root system with degree 5 or greater can't be solved algebraically. First integrals for differential equations are functions which remain constant along a solution to that differential equation. So for 8 solutions there is some eight order polynomial p8(x) = ^=1(x — Xn), with 8 distinct rootsXn that are constant along the 8 solutions. Since p8(x) = p5(x)p3(x), a branch of algebra called Galois theory tells us that fifth order polynomials have no general algebraic system for finding its roots, or a set of solutions that are algebraic. This means that any system of degree higher than four are not in general algebraic. At the root of the N-body problem Galois theory tells us there is no algebraic solution for N > 3. Galois theory is a subject outside the scope of this book, but it is the basis of abstract algebra, groups and algebras used in physics.

This reasoning is the basis for a series of theorems by H. Bruns in 1887, which lead to Poincare's proof that for n > 2 the n-body problem is not integrable. For this reason the stability of the solar system can't be demonstrated in a closed form solution, but only demonstrated for some finite time by perturbation methods. In chapter 14 the results of numerical integrations of a reduced solar system with the sun, Earth and Jupiter are illustrated. The random nature of this dynamical system is illustrated, and reduced to a simple model of a random chain, or Ising model. This is then used to estimate whether known extra-solar systems might support a planet in an orbit sufficiently stable to permit the evolution of life.

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