European Solar Polar Orbiter Mission

Approximate Trip Times

Initial, approximate trip times can be obtained by adding the trip times for each of the phases. This however neglects the phasing of the orbits, the Earth ephemeris, and the orientation of the line of nodes of the polar orbit. Note that the line of nodes requirement means that a minimum trip-time opportunity occurs every six months. The launch window can however be considered open at all times through the year, with a trip-time penalty incurred as launch varies from the two optimal start epochs. In general the actual trip-time will be slightly longer than these approximate times, as will be seen later.
It is seen in Figure 7 that the cranking time is very sensitive to the cranking orbit radius, while the curve levels off at higher accelerations. The closer the orbit is to the Sun, the faster the inclination changes. The rate of change of inclination is however constant per integer number of orbit revolutions and is independent of orbit radius, assuming a circular orbit.Error: Reference source not found Low radius orbits have shorter periods and so the inclination change is effected more rapidly. For a launch C₃ of zero, optimising circular-coplanar transfers from 1 AU to the polar/cranking orbit radius using NPSOL produced inward spiral trip-times. The spiral times were added to the cranking times at the resonant orbit radii and the total 2-phase trip time was found. Cranking at 1 AU or 0.63 AU is tremendously time consuming, thus a 3-phase approach must be adopted to reach these orbits. Further, a 3-phase approach is beneficial for the higher resonance numbers if the close solar orbit thermal loads can be withstood, as seen in Figure 7. Figure 7 concurs with the 0.5 mm s^-2, 2-phase transfer to 0.48 AU polar orbit that was previously generated and found to have a duration of order 5 years.Error: Reference source not found^{, Error: Reference source not found} It should be noted here that past work does not include positive C₃ launches or take into account Earth ephemeris, orbital orientation and phasing, which will be included later in this paper. A moderate characteristic acceleration of 0.5 mm s^-2 would require a close cranking orbit of 0.3 AU for a mission duration below 5 years to a N = 1 orbit. Using a positive launch C₃ could alleviate this requirement.

Optimized Inward Spirals to Circular Low Inclination orbits at 0.48 AU

In the parametric study conducted by Sauer the inward spiral is optimized to 15 deg inclination before starting the cranking maneuver. If a third outward spiral phase was required then the final 15 deg to reach polar orbit was also optimized. Sauer utilizes a locally optimal inclination control law, which has the drawback of making the generation of fully phased orbits very difficult. It was also noted that the total transfer time is relatively insensitive to initial cranking orbit inclinations above 10 deg.Error: Reference source not found This paper concentrates on the two-phase transfer to a 0.48 AU polar orbit. The effect of optimizing a circle-to-circle inward spiral from 1 AU to 0.48 AU was investigated, for a number of different target orbit inclinations. The optimized inward spiral times were added to the remaining cranking time necessary to match the inclination to polar orbit. The NPSOL optimization matched the semi-major axis, eccentricity and inclination as constraints. It was found that although the overall saving is less than 6 months, optimizing to 10 – 20 deg is significantly better than for 5 deg. In general, 15 deg seems the optimum value, as was found by Sauer.Error: Reference source not found As the initial cranking orbit inclination is increased beyond these values, many more revolutions are needed and so the optimizer requires more control nodes – placing greater demands on the optimizer.

Figure 7 Three-phase total trip times to N=1, 2, 3 solar polar orbits

Utilizing Excess Launch Energy

Circular-coplanar optimizations of the inward spiral phase were conducted while increasing the launch C₃ to 40 km² s^-2, in opposition to the velocity of the Earth. We note that non-zero launch declinations were found to have a significantly adverse effect on sail transfer times and are thus not utilized. 51 control nodes were used. Figure 8 shows the effect of using positive C₃ on the spiral-down time to a 0.48 AU circular orbit over a range of characteristic accelerations. For higher accelerations and higher cranking orbit radii it was found that the curve levels off sooner than for low accelerations and low cranking orbits. There is therefore an increased benefit in using the excess C₃ capability to reach lower cranking orbit radii with low performance sails.

Figure 8 Spiral-in time to circular cranking orbit at a radius of 0.48 AU for characteristic accelerations of 0.3, 0.4 and 0.5 mm s^-2, against launch C₃.

We recall that a characteristic acceleration of 0.5 mm s^-2 was designated earlier within the approximate analysis to reach a 0.48 AU polar orbit with a total transfer time of order 5 years. An attempt was made to reduce the sail performance requirements by using the excess C₃ available, with optimized spiral-in to 15-20 deg inclination. In order to produce an approximate SPO transfer with a trip time of order 5 years, it was found that the characteristic acceleration needed was of order 0.42 mm s^-2 for a C₃ of 40 km² s^-2. We note that the Soyuz Fregat 2-1b from Kourou has a minimum launch mass of 620 kg, corresponding to a zero declination positive C₃ of 38.8 km² s^-2, thus we anticipate that the actual trip time will be slightly in excess of 5 years.