Direct Transfer with Lambert Solver

Lambert Solver

We saw in the Hohmann transfer section that a Hohmann trasfer between Earth and Neptune will take just over 30 years. This is too long for a realistic mission. We need a higher energy transfer trajectory. The higher energy will mean that the spacecraft must use more delta-V to get to Neptune, but it will arrive at Neptune sooner. To design this new trajectory, we will use a Universal Variables Lambert solver.

Lambert's problem is; given two position vectors and the time of flight between them, find the trajectory that connects them. Solving for the velocity at each position vector is enough to define the connecting orbit. There are a variety of solution methods to this problem. For this project, a Universal Variables solver was written, following Algorithm 58 in [D. A. Vallado]. The position vectors will be the positions of Earth and Neptune at given times, and by varying the time of flight between them we can create a number of trajectories, some of which may be more optimal than others.

Solution Method

In this portion of the project, we will use the UV Lambert solver to construct a family of trajectories from Earth to Neptune. The c3 parameter will be used as the primary selection criteria. c3 is proportional to energy, and is frequently used as a criteria for selecting launch vehicles.

We would like to minimize c3.

Additionally, we will take January 1, 2017 through December 31, 2017 as the launch window for the spacecraft. The arrival window at Neptune will be January 1, 2022 through December 31, 2027, meaning that the transfer time of flight will be 5-10 years. It was found through trial and error that the computation time on the author's computer became prohibitive for much longer trajectories, and shorter trajectories were too high energy to be practical.

Initial positions and velocities of both planets on January 1, 2017 were taken from JPL's HORIZONS system.

These were converted to orbital elements, and then propagated forward using the same technique we have been using most of the semester. The positions of the planets were found for each launch day and arrival day combination in their respective windows. The UV Lambert solver was run using the planet positions and time of flights, to generate the possible trajectories.

Finally, only Type I transfers were considered. Type I transfers are also known as the "short way" solutions to Lambert's problem, and Type II are the "long way." This selection was made for the sake of computation time.

Direct Transfer Results

The resulting porkchop plot of possible Earth to Neptune trajectories is shown below. It shows contours of c3 as well as the hyperbolic excess velocity at Neptune arrival.

Some features of this plot are worth noting. First, the ideal launch window seems to be around 125 days after Jan. 1. c3 values will be large for any direct transfer to Neptune, but launching earlier than ~75 days or later than ~175 results in prohibitively high values. Second, the smallest values for c3 and arrival excess velocity occur at the top of the plot for an arrival 10 years after January 1, 2017. This makes sense, as longer transfers mean that the trajectory approaches the ideal, lower energy but longer, Hohmann solution.

With the porkchop plot generated, we can find the launch and arrival day combination that results in the smallest c3. These day, and the associated parameters, are summarized below:

Unsurprisingly, it this solution utilizes the entire 10 year arrival window. With these values, we can calculate the delta-V necessary to put the spacecraft on the transfer trajectory and to insert into orbit around Neptune. We will use the same 200 km parking orbit around Earth, and target the same 100000 km orbit around Neptune. The calculation is then done via the same method used in the Hohmann transfer section using the newly calculated hyperbolic excess velocities, and results in:

The delta-V to transfer from the Neptune orbit to Triton, and then to insert into a Triton orbit, will be exactly the same as before, resulting a total delta-V of the direct transfer of: