Sequence Evaluation

This step assesses the feasibility (from an energetic point of view) of each sequence, and gives an approximated value of the relative velocity at the arrival planet. Since this quantity is often important when dealing with multi gravity assist transfers, it can also be used to rank each sequence in the list.

Each of the sequences found at the previous step is evaluated: since the number of sequences can still be pretty high, the idea here is to have a fast assessment of each sequence, using a reduced model, which can be considered an extension to what is proposed in [74]. In particular, the following assumptions were adopted:

 Orbits of all the bodies at which swing-by can be performed, and the spacecraft departure body, are considered circular;

 All the orbits and transfers are considered to lie in the same plane (planar system);

 No phasing is taken into account: a swing-by or rendezvous is possible every time the orbit of the spacecraft intercepts the orbit of the body;

 No other propelled manoeuvres are considered, other than the launch.

Swing-bys are responsible for changing the heliocentric energy of the spacecraft; this makes this model unsuitable for evaluating the change in relative velocity in resonant swing-bys;

 No overturning of the outgoing heliocentric velocity vector after swing-by is allowed. This means that, if the rotation of the incoming relative velocity vector leads to an outgoing relative velocity vector which is on the other semi-plane with respect to the planet velocity, then the rotation which gives the maximum acceleration or deceleration of the spacecraft in the heliocentric reference is considered (see Fig. 3.14 and Fig. 3.15).

Fig. 3.14 and Fig. 3.15 show the triangles of velocity for a planar swing-by with a planet with circular orbit. The velocity of the planet is purely transversal ( ). The velocity vectors and angles referring to the incoming conditions at infinity before the swing-by are denoted by a superscript (–), while those after the swing-by are identified by (+).

vP

ˆθ

Fig. 3.14. Velocity triangles for a swing-by of a body with circular orbit (its orbital velocity is purely tangential).

Fig. 3.15. Deflection correction in order to avoid overturning. Primed angles and dotted lines refer to the case in which overturning occurs.

The subscript (1) or (2) refer to the two possible outcomes of the swing-by, depending on the direction of deflection (see Section 2.2.2). In the case represented in Fig. 3.14, there is no possible overturning of the velocity vector: in fact, the deflection ₂ originates the maximum increase in outgoing absolute velocity . In the same way, the minimum outgoing velocity

v2^

v1^ is obtained with deflection ₂. In Fig. 3.15, instead, the maximum deviation ₂ does not provide the maximum magnitude of the outgoing velocity vector. In fact, that is achieved by using a smaller deflection ₂. This last one is the one used for computations within this model, to avoid overturning.

Given this procedure, and a sequence which is to be assessed, the trajectory model is used in the following way: depending on the problem, there can be a launch, or the initial conditions for the spacecraft are given at a certain planet. In the case of launch, the spacecraft is assumed to be at the departure body (thus on a circular orbit). The time of launch is not influent, as body’s orbits are circular and

^

no phasing is considered. The launch excess velocity is tangential (to have the maximum or minimum variation of semi-major axis), and with the same verse of the velocity of the planet if the next planet in the sequence has a longer radius, otherwise is on the opposite verse. Instead, in case the initial velocity is given, then no other calculations or assumptions on launch are needed.

The resulting orbit (either after launch or obtained with the assigned initial velocity) is subsequently computed: if this orbit intercepts the orbit of the second body in the sequence, then a swing-by of that planet can be exploited. Assuming that the planet is in the correct position regardless of the arrival time, the incoming relative velocity is computed.

SWING-BY

Given the incoming conditions, a planar swing-by is fully determined by the pericentre and the direction of deflection. Petropoulos et al. [74] proposed to use the lowest allowed altitude for all the swing-bys, in order to maximise the heliocentric velocity variation (and thus maximising the change in energy). Still, fixing a particular value for the closest approach, there are two possible outgoing heliocentric velocities: one is higher (positive turning of velocity vector) and the other one is lower (negative turning) than the incoming velocity. The same authors made the following choice: if the orbit of the next different planet in the sequence has a longer semi-major axis, then the positive turning is chosen, otherwise the negative turning is considered.

This choice seems reasonable when assessing the feasibility of a given sequence, which is what is done in the work of Petropoulos. On the other hand, we think that this assumption is not suitable when an estimation of the final relative velocity is required. In fact, using the lowest possible altitude for the swing-by enables to determine whether it is possible to reach a given radius exploiting the swing-bys only, from an energetic point of view (i.e., neglecting plane change and phasing). Under these assumptions, though, the final relative velocity is just one of the many possible final relative velocities which can be achieved by changing the altitudes of the various swing-bys. If the problem requires to reach the final planet with the lowest possible relative velocity (a very common objective), or more seldom with a specific value of relative velocity, then this way does not provide any information on which sequence is possibly the best.

For this work, it was decided that several choices for the radius of the pericentre of the swing-by hyperbola have to be assessed, in order to have an estimation of the range of the final relative velocity which is achievable with the considered sequence. Certainly, it is not possible to consider a continuous set of values for the radius of pericentre, for each swing-by, as each combination of radii leads to a different trajectory.

Let be the i^th possible radius (expressed in radii of the planet). If the sequence under evaluation involves

,

rp i

n swing-bys, and the list of possible radii sb

contains k elements, then the number of combinations of radii is k . Considering ^nsb that each swing-by with a fixed altitude has two possible outgoing velocities

(depending on the direction of the rotation of the relative velocity), then the total number of trajectories to assess for each sequence is

 

Despite the number of trajectories to evaluate can grow considerably with the number of swing-bys and number of possible radii, the evaluation of each trajectory is very fast, as this model does not imply the use of any computationally expensive algorithm (Lambert or Keplerian propagation), but the whole problem is solved analytically.

FEASIBILITY

Having fixed the combination of radii, the new orbit after the swing-by is computed and the algorithm continues looking for the intersection with the following body’s orbit of the sequence. If the swing-bys allow the spacecraft to reach the last planet in the sequence, then we say that the sequence is energetically feasible, and the velocity relative to the last planet (v_) is computed and stored.

The procedure is repeated for each combination of radii/direction of deflection, and the minimum value of (or the one closest to a target value) is stored. This value will be used as an indicator of the possible achievable value of v .

v_

