Approximate closed-form solution for solar sail spiral trajectories with sail degradation

Solar sails have long been considered as a means of enabling new high-energy missions, such as solar polar orbiter, planetary sample return and heliopause probes, along with families of highly non-Keplerian orbits for space weather and Earth observation missions, as reviewed in Refs 1 and 2. While most prior
analysis of solar sail orbital dynamics assumes that the optical properties
of the sail are time-invariant, it is expected that the sail membrane will slowly degrade due to cumulative, long-term exposure to solar radiation. The effect of such degradation was investigated in some detail by Dachwald and co-workers through a combination of modelling and numerical optimization. 3,4. As expected, it was found that degradation has an impact on the trip time
of solar sails to target orbits and the ability of solar sails to realize non-Keplerian orbits. In this Note, an approximate closed-form solution is presented for solar sail spiral trajectories with sail degradation. Since exposure to the space environment is cumulative, the impact of degradation on the sail thrust magnitude forms an integral function over the spiral duration. The time evolution of the solar sail spiral trajectory is therefore described by an integro-differential equation, which for this problem does not apparently possess an explicit closed-form solution. However, it is possible to obtain an implicit
solution to the problem which provides an approximate representation of the
evolution of the solar sail spiral trajectory. This allows an initial estimation of the impact of sail degradation on mission performance to be made.
Limits are also found which bound the motion of the solar sail to an annulus, whose inner and outer radii are defined by the sail degradation rate. These limits correspond to the asymptotic behaviour of the solar sail where the sail
becomes completely absorbing, the transverse component of sail thrust vanishes and the spiral terminates. In addition, it is shown that the optimal fixed sail pitch angle required to maximize the inner and outer radius of the annulus differs from the usual sail pitch angle which maximizes the transverse component of sail thrust. For a solar sail subject to degradation, the optimum fixed sail pitch angle must trade-off maximising the transverse component of sail thrust while minimising the projected sail area exposed to solar radiation.
Finally, while these asymptotic limits of motion represent an operational constraint, they could in principle be used as a means of enabling entirely
passive orbit transfer. For example, a small, low cost solar sail with a passively
fixed pitch angle could deliver a payload onto a quasi-circular spiral
trajectory. If the sail film is engineered to degrade at a certain rate, the asymptotic orbit the sail winds onto can then be chosen a priori. The solar sail will then perform a near circle-to-circle transfer, in principle satisfying a simple two-point boundary value problem, but using only a passively fixed sail pitch angle. Such a mode of transfer could be used to deliver payloads to given
heliocentric orbits, for example for space physics applications.