A Quick Estimate

A few days ago I was invited to take part in a mock architecture study looking at Deep Space CubeSat missions. As a multi-body dynamics fanboy, my heart jumped at the possibility of using the interplanetary superhighway to preserve our meager delta-V budget. The downside of using the interplanetary superhighway is that it is very slow. I looked around hoping to find a design rule that I could use to explore the trade-off between the delta-V savings and the time of flight increase. When I couldn’t find one, I wondered if I could create a simple design rule. Unfortunately, n-body dynamics are hard, so I was only able to create a the following small design rules of thumb for transferring from the primary to the secondary in a 3-body system with a mass ratio of about μ=0.0123, which is the Earth-Moon system.

A trajectory with a 10x increase in Time of Flight reduces the delta-V to 75% that of the optimal lambert trajectory (Power Law Version)

A trajectory with a 15x increase in Time of Flight reduces the delta-V to 70% that of the optimal lambert trajectory (Conservative version)

Reducing the delta-V to below that of 65% of the optimal lambert trajectory will require an increase of over 75x in Time of Flight

What is the Interplanetary Superhighway?

Lagrange points are points in space in a 3-body system where a spacecraft would remain frozen between the two main bodies. Every 3 body system, from the Earth-Moon, Sun-Earth, to even the Pluto-Charon system has 5 of these Lagrange points. Imagine threads leading away from each systems Lagrange point and cycling through space. Sometimes these threads loop back to the Lagrange points and other times lead to different Lagrange points. By riding these threads, and sometimes using a little fuel to jump between different threads, we can go vast distances across our solar system while using very little fuel. The two big downsides to these trajectories are that they can be really hard to find, and they take an extremely long time to go places. I’ve simplified this down a lot, but in my opinion, the best textbook on this topic is Dynamical Systems, the Three-Body Problem, and Space Mission Design.

How was the design rule created?

Dynamical Systems, the Three-Body Problem, and Space Mission Design, a textbook by Dr.s Koon, Lo, Mardsen, and Ross, has on page 285 a figure describing both the delta-V requirements as well as the TOF for different Earth to Moon trajectories. I extracted the data from this figure using the web plot digitizer into a CSV. I then fitted a power law to the data recovering the following relationship between delta-V and TOF.

Non-dimensionalizing the variables using the standard characteristic quantities used in the 3 body problem we obtain the new relationship

We can then obtain the rules of thumb posted in the intro by plugging in a 10% increase in TOF. The conservative estimate in the second bullet point comes from the first datapoint in the figure.

Does this Design Rule cover the Interplanetary Superhighway?

No, not really. When people talk about using interplanetary superhighway, they normally assume that you start near or on an “on-ramp” and you’re going to a point also near an “off ramp”. This is done because the delta-V required to go between these points is minimal and it is in these situations that the interplanetary superhighway shines. In this case we are going from a high earth orbit to a stable lunar orbit.

Is this Rule U seful?

Not really.

In reality the patched conic trajectories and the 3-body dynamics trajectories belong to different families so there likely isn’t a smooth transition between the two. On the other hand, this rule did provide a fun example to use the web-plot digitizer, talk about the interplanetary super highway, and link to Dynamical Systems, the Three-Body Problem, and Space Mission Design.

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