The design of trajectories for spacecraft travelling from one body (usually the Earth) to another has been extensively studied since well before the first actual space missions; Space Mission Analysis and Design provides a good introduction to the subject. Describing such a trajectory is inherently more complex than modelling the orbit of a spacecraft around the Earth, as the effects of the Earth, the Sun and the target body all have to be considered. In practice, matters are simplified (at least for initial trajectory planning) by using a patched conic approximation in which the spacecraft is assumed to be acting under the influence of only the body exerting the strongest gravitational influence on it. For simple orbits such an approach is adequate for estimating time-of-flight and the delta-V required at different phases of the mission.

For most interplanetary missions the spacecraft trajectory must meet a number of demanding requirements, the most important of which is the minimisation of the delta-V required during the mission. A high delta-V requirement translates into either a smaller payload or a heavier spacecraft, and thus heavier launch vehicle. Also, for missions landing on, or going into orbit around, a target it is desirable to minimise the arrival velocity as this reduces the braking fuel requirement and/or the size of the entry heat shield. The effect of these demands is usually to constrain interplanetary probes to minimum-energy, long-duration Hohmann transfer orbits. As well as involving prolonged cruise phases, these orbits require launch during periodic windows of only a few days or weeks duration at intervals of one to two years. In extreme cases (usually associated with traditional-design large probes) the requirement to keep the mission delta-V to within the capacity of an affordable launch system has lead to multiple-encounter gravity-assist trajectories being used. The Galileo mission used a VEEGA trajectory, i.e. a deep-space manoeuvre, Venus flyby and two Earth flybys in order to reach Jupiter after a total flight time of 6 years.

For a flyby mission such as involved in this project though, the requirements are less stringent. Although it is always desirable to keep flyby velocity low so as to prolong the encounter, the relative velocities involved in NEO asteroid encounters (see Section 6) are high enough that not much can practically be done to reduce them whatever trajectory is used. As such the constraint of minimising encounter velocity does not apply.

The parameters of this mission allow trajectory planning to be simplified still further. There will only be a narrow time window around the period of minimum range during which the encounter can take place, and it is desirable that the probe reaches the encounter point as quickly as possible. Minimising flight time rather than delta-V thus becomes the important concern. Indeed, initial modelling of the probe's trajectory showed that rather than trying to set up a particular transfer orbit (the usual case for interplanetary flights) the actual aim was to reach a specified point in space as quickly as possible. The probe's short mission life meant that its actual orbit and thus subsequent trajectory were not important, although in the context of any extended mission they would be of interest.

So, the question arises: How fast can we reach an intercept point 0.05 AU (7.5 x 10⁶ km) from Earth? Any interplanetary flight involves giving the probe enough energy to escape the Earth's gravitational potential 'well' on a hyperbolic orbit, i.e. one where its kinetic energy exceeds (at any point on the trajectory) its gravitational potential energy. The latter being taken as negative tending to zero at infinite distance from Earth; at infinity the excess energy is seen as a hyperbolic excess velocity towards which the probe's speed asymptotically reduces as it recedes from Earth. In practice, the hyperbolic excess velocity (V_hyp) is attained relatively soon after injection into a hyperbolic trajectory.

If a probe is given injection velocity V_inj at a radius r_inj from the centre of the Earth then its specific energy eta (energy per unit mass) is given by the vis-viva equation:

(7-1)

where mu is GM_Earth, the gravitational parameter of the Earth. If is positive then the probe is in a hyperbolic orbit and the final V_hyp will be related to eta by*:

(7-2)

[*Many astronautical references quote eta in (km/s)² rather than V_hyp; in this case it is usually referred to as C3.]

For a typical injection altitude of 200 km (i.e. r_inj = 6578 km) Fig 7.1 shows the variation of V_inj:

Note that a small increase in V_inj can give a big increase in V_hyp, particularly at velocities just above the critical 'escape velocity' where becomes positive. This is more noticeable if we plot V_hyp against the delta-V from a circular orbit at radius r_inj:

It is evident that a delta-V of less than 4.5 km/s from a circular parking orbit - equivalent to a delta-V of only 1km/s above that required to reach escape velocity - will give a V_hyp of approximately 5 km/s. As modelling the performance of various candidate launch options indicated that this was a feasible injection velocity (see Section 10) then 5 km/s was chosen as a nominal V_hyp.

A V_hyp of 5 km/s allows a wide range of potential targets to be reached, well beyond the required 0.05 AU required range. Indeed, depending on the direction of the trajectory asymptote**, a probe departing Earth with a hyperbolic excess of 5 km/s can end up in heliocentric orbits very different from Earth's, as shown in Table 7.1. This table shows the heliocentric elements calculated for a probe moving at 5 km/s away from the Earth. Four cases are shown; two with the velocity asymptote parallel to the Earth's motion around the Sun (one prograde, one retrograde), one with it radially outwards and one normal to the Earth's orbit.

[**The trajectory asymptote is the linear (relative to the Earth or other primary body) trajectory that a spacecraft on a hyperbolic orbit asymptotically approaches as its distance from that body increases.]

As Table 7.1 shows, a probe leaving Earth with a V_hyp of 5 km/s is likely to enter a significantly different orbit than Earth around the Sun. It thus seemed a reasonable assumption that the probe would leave the near-Earth region very rapidly; if this was true, then the probe's trajectory out to the maximum encounter distance of 0.05 AU could be approximated as a straight line at V_hyp.

To verify the validity of this assumption the orbit of a probe on such a trajectory was propagated out to a distance of 0.05 AU from Earth. This was done by calculating the heliocentric elements of such a probe in the same manner as was done for Table 7.1. These elements were then used, via Kepler's equation for motion in an elliptical orbit to calculate the probe's position vector relative to the Sun. A parallel calculation of the Earth's position in its own orbit allowed the position vector of the probe relative to the Earth to be obtained. By doing this at intervals of a few days a plot of the probe's path in an Earth-centred reference frame could be obtained.

Fig 7.3. Trajectory relative to Earth of a probe departing with V_hyp = 5 km/s in the direction of Earth's motion around the Sun. Vector from Sun to Earth at time of departure is along 0°. Time ticks at 3-day intervals.

Fig 7.3 shows a polar plot of the probe's position relative to the Earth for a trajectory with a V_hyp of 5 km/s along the direction of the Earth's orbital motion. The time ticks are intervals of 3 days and the scale is in AU. The 0° baseline is in the direction radially outwards from the Sun at the time of launch. It can be seen that in this case the trajectory is effectively a straight-line, constant velocity path. This is in considerable contrast to most deep space trajectories, which are typically transfer orbits where the probe's position and velocity vectors relative to both the Earth and the target body very considerably over the course of the mission.

Fig 7.4 shows the results of the same trajectory calculation repeated for the case where V_hyp is 5 km/s radially outwards from the Earth relative to the Sun. Again, the trajectory relative to the Earth is effectively a constant-velocity straight line over the area of interest.

Fig 7.4. Trajectory relative to Earth of a probe departing from Earth with V_hyp = 5 km/s radially outwards from the Sun. Sun-Earth vector at time of departure is along 0°. Time ticks at 3-day intervals.

When the probe's path is propagated much beyond the near-Earth region though, this linear approximation becomes invalid. For example, Fig 7.5 shows the probe's trajectory relative to the Earth for the same injection conditions as Fig 7.4, but out to a distance of 0.6 AU. The time ticks in this instance are at intervals of 3 weeks.

Fig 7.5. Trajectory relative to Earth of a probe departing from Earth with V_hyp = 5 km/s radially outwards from the Sun. Sun-Earth vector at time of departure is along 0°. Time ticks at 3-week intervals.

For encounter distances of up to approximately 0.05 AU though, injection into an intercept trajectory with V_hyp of 5 km/s gives a short (circa 20 day) flight time and a simple, near straight-line flightpath. This last point significantly simplifies other elements of mission design such as estimating the delta-V requirement for trajectory correction. It also allows the assumption that the probe maintains a constant orientation with respect to both the Sun and Earth during the cruise phase, simplifying power, thermal and communications payload design.

Trajectory Correction

In Section 10 it is estimated that for a solid propellant boost stage the 3-sigma error in injected velocity would be 85 m/s. To return the probe to its correct trajectory an initial trajectory correction manoeuvre of up to 85 m/s will thus be required. This will not, though, remove all errors and subsequent corrective burns will be needed. For the purposes of the flyby calculations it has been assumed that the probe position is known relative to the target to within about 1.5 km, so the trajectory to the intercept point must be accurate to this extent.

If the first trajectory correction manoeuvre (TCM) corrected all errors then the required TCM delta-V would be 85 m/s. A simple model of TCM strategy would be to assume that all TCMs were in error by some amount, say 10%. If this was the case then the TCM delta-V budget would be (1+0.1+0.01…) x 85 m/s, i.e. about 95 m/s. However this assumes that all TCMs are carried out one after the other. In fact time will be needed to track the probe to determine any velocity error; this will delay each subsequent TCM and require it to be larger in order to achieve the appropriate correction. As a first-order approximation, it was assumed that the total TCM delta-V budget would be twice the initial TCM delta-V, i.e. a total of 170 m/s. This will be provided by the probe's reaction control system and so needs to be included in its RCS fuel budget. The number of TCMs was estimated by again assuming a 10% accuracy and working out how many such burns would be needed to reduce the initial trajectory error to that required at the flyby. For this mission profile, the error after injection would be that resulting from an 85 m/s velocity error for 1.5 x 10⁶ s, i.e. 127,500 km. To reduce this to 1.5 km will require 5 successive TCMs if each is accurate to within 10%.

Summary

Injection of the probe into a heliocentric orbit with V_hyp relative to Earth of 5 km/s results in it following an essentially direct trajectory to the target out to a distance of 0.05 AU. For an injection velocity error of 85 m/s (the estimated 3-sigma error for the boost stage being used) it is estimated that up to 5 trajectory correction manoeuvres will be required with a total delta-V budget of 170 m/s.