About a year ago, I wrote a review of an AAS conference paper that I coauthored with a few of my astrogator friends, Mike Loucks and John Carrico regarding an mission design tool for enabling the use of LEO depots for deep-space missions. At this year’s AAS/AISS Astrodynamics Specialist Conference in Snowbird, Utah, we did a follow-on paper, with the help of Altius’s Matt Isakowitz Fellow, Brian Hardy, and I wanted to provide a review of this paper, since it was a lot of fun, and I think extremely relevant and timely. As with last time, the paper will be published in a future volume of Advances in the Astronautical Sciences^{1}.

Before I review the paper, here’s a full-text copy for reference: AAS 18-447: RAAN-Agnostic 3-Burn Departure Methodology for Deep Space Missions from LEO Depots

**Backstory/Introduction**

As a quick reminder of what led us to develop these mission planning techniques, or for those who haven’t had a chance to read the previous blog post, back in 2011 when there was a lot of NASA interest in orbital propellant depots, some flight dynamicists at NASA Johnson Space Center raised a serious concern about the feasibility of using LEO propellant depots for deep space missions. The tl;dr version of this argument is that for any given interplanetary departure, you have to leave along a certain V-infinity vector, and for a reusable LEO depot that wasn’t just launched for this specific mission, the odds that the depot plane would align with that V-infinity vector at the right time was small. You could launch a depot per-aligned for one specific mission, but the odds of it then lining-up correctly for any particular future opportunity was small enough (<25%) to make LEO depots impractical.

What we did was come up with a 3-burn departure that would allow you to leave a LEO depot into a phasing/alignment orbit that would put you back at perigee, in the right place, at the right time, and with the right alignment to do your planetary injection burn, even if the depot’s plane wasn’t aligned with the departure vector at the departure time. In fact, we found that in many cases it was possible to mount a deep space mission from a depot even if the depot plane never intersects with the V-infinity vector (i.e. if the declination^{2} of the departure asymptote^{3} is higher than the inclination of your propellant depot’s orbit), so long as it’s close enough. What this means is that you could have a LEO depot that you refill and reuse multiple times for a wide range of missions without having to move the depot around to line things up for a given mission. Which is kind of important for a depot to be economically useful.

In our first AAS paper, we described the genesis of the 3-burn methodology, which was actually a paper by Selenian Boondocks alumni Kirk Sorensen, and showed how it could be used to enable a Mars mission or a mission to a NEO with a very high declination angle (2007 XB23). However, to simplify things for the first paper, we assumed a phasing orbit with a specific apogee altitude, which basically still required you to align the depot plane with that phasing orbit, which kind of defeats the purpose. We knew we could use this technique for enabling departures from a depot regardless of what its RAAN^{4} was at the time of the departure window by varying the altitude of the phasing loop, but we hadn’t been able to take things that far by the time we had to present last year’s paper.

So the purpose of this paper was to flesh-out the methodology showing how you could use it for missions regardless of where the depot plane was at the desired departure time. Also, to illustrate how powerful this capability was, we illustrated the use of this RAAN-agnostic 3-burn maneuver for enabling a rapid-fire series of deep-space missions from a single LEO depot–4 planets, 1 moon, and 4 NEOs in a 5 month timeframe. Without further ado, I’ll dive into the work we did in this paper.

**Methodology Refinement**

We described the methodology in a more rigorous manner in the paper, but here’s a quick summary:

- Identify the desired departure geometry (C3, declination and RAAN of the departure asymptote, the resulting locus of periapses
^{5}, and departure date), and determine the orbital parameters of your depot at around the time of your planned departure. - Check if a simple one-burn departure is possible–the odds aren’t great, but if the plane happens to be lined-up correctly, may as well keep things simple.
- Calculate when to enter the phasing orbit–if your depot isn’t aligned with the departure asymptote at the departure date, you need to enter a phasing orbit the
*last time*your depot was optimally aligned. Because your depot plane precesses over time, you can time-step back to the last time you were aligned properly, and have that be the time you do the injection burn to enter your highly elliptical phasing orbit. - Design your phasing orbit–first you calculate how long you need to be in the phasing orbit, and then you can pick a one, two, three, or four loop phasing orbit, with the loops taking some integer fraction of the required phasing time. Lastly, using a high-fidelity simulator you will want to add in required plane changes and/or perturbation correction burns at the apogees of the phasing orbits.
- Calculate the final departure burn and tally the required Delta-Vs for each of the maneuvers.

While for the mission simulations we did in the paper we mostly eyeballed several of the steps and then used targeting algorithms to correct for eyeballing-errors, it should be possible to automate these steps^{6}.

In the process of designing this methodology and exercising it, we learned several lessons worth mentioning (in no particular order other than what I could think of when writing this summary):

- If the declination of the departure asymptote is lower than your depot inclination, the lowest delta-V departure will happen if you enter your phasing orbit the last time the depot plane intersects the departure asymptote
^{7}prior to the departure date. In this case, you don’t have to do a plane change to align for the departure, just corrections for lunar or solar perturbations. - If the declination is higher than your depot’s inclination, but the angular extent of the locus of periapses
^{8}is larger than the difference between the two (ie if your depot plane at any point crosses through the locus of periapses), you can still use this 3-burn departure methodology, you’ll just have to do a plane change at apogee to align your final departure plane with the departure asymptote. Since this plane change takes place at near escape velocity, the cost of the plane change can be very modest. The delta-V optimal timing for this orbit would be at the last time where the depots orbital plane came closest to intersecting with the departure asymptote^{9}. - The angular extent of the locus of periapses is a function of the injection C3. The faster you have to leave the earth, the wider that locus is. So for a medium-inclination depot (such as one in an ISS-coorbital plane), the only missions you can’t use the 3-burn departure method for are a few NEO missions with high declinations but very low C3. Those are fairly rare, and there may be more complicated departure methodologies that can enable these, but one brute-force solution would be to have a small depot in a near-polar orbit.
- For either case, the solution with the lowest total trip time (including phasing orbit) will occur if you enter your phasing orbit the last time the locus of periapses intersects your depot orbital plane prior to your departure date
^{10}. In this case you’ll definitely need a plane change at apogee. - As mentioned previously, phasing orbits don’t have to be a single-loop. You can actually go for anywhere from 1-4 orbits while still keeping the orbit elliptical enough to freeze your plane’s orbital precession.
- Phasing orbits with several smaller loops tend to be less susceptible to solar or lunar perturbations, which will vary in magnitude depending strongly on where the moon is relative to your departure asymptote and your phasing orbit
^{11}. On the other hand, with smaller numbers of phasing loops, more of the departure burn is performed by the refueled upper stage, which typically is higher performance than the kick stage(s). Long story short, you’ll want to check the 1, 2, 3, and 4 phasing loop options to see which is performance optimal for a given mission, because it’ll vary. - Worst case trip-time penalties that we saw were less than 45 days. For a robotic mission, this is probably not an issue, but for a human spaceflight mission, these could be an annoying penalty. One way to solve this would be to use a 3 or 4 loop phasing orbit, and use the depot to fuel and launch everything in the departure stack other than the crew, and then have the crew launched separately only during the last phasing loop, meaning you could keep the trip-time penalty for the crew below ~10 days, and only add two extra Van Allen Belt crossings, at the expense of requiring a launcher that can send the crew capsule into the same highly-elliptical phasing orbit as the mission stack
^{12}.

I’m going to take a break at this point to keep the blog post from getting too long. In the second half of this review, I’ll go over the Interplanetary Blitz campaign I mentioned in the introduction.

#### Jonathan Goff

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- I’ll update this post with the volume number once I have it
- The angle between that departure vector and the earth’s equatorial plane
- The more technical term for the V-infinity vector
- Right Angle of the Ascending Node, which is a fancy way of describing where an orbit crosses the equator going northward relative to the earth’s prime meridian
- A ring of potential perigees that you could do your final injection burn from at the departure date. This ring is centered on the departure asymptote vector but on the far side of the planet from the departure vector, ie the anti-asymptote side. The goal of this whole 3-burn maneuver is to get your spacecraft to a point on that ring, with the plane of its orbit intersecting the departure asymptote vector at the departure date.
- Which would be a fun Master’s thesis or PhD dissertation topic for an interested graduate student! Not everyone has the luxury of having astrogator friends who’ve done this enough times professionally to just be able to eyeball things close enough for a targeting algorithm to finish the job.
- Mathematically, you can find this point by finding when the dot product of your depot’s orbit angular velocity vector and the departure asymptote is zero.
- The angle between the departure asymptote line and the vector between the earth’s center and that ring of injection points.
- Mathematically, this would be the point where the dot product of the depot orbit angular velocity vector and the departure asymptote are at a minimum
- I’m pretty sure there’s a closed-form mathematical way to describe this, but I’m not even going to pretend like I’m competent enough to derive that equation!
- It’s worth noting that in theory, those perturbations can sometimes actually help!
- This would probably be possible with Dragon V2 on a Falcon Heavy, with Starliner on a Vulcan/Centaur with lots of solids, or with Orion on SLS

I hope this will retire some of the doubt about fuel depots in LEO

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I believe I know one of the dynamicists you’re talking about. This is work I’ve wanted to do for a long time. Thank you. Can’t wait to read about it!

While it should retire doubt, it is quite difficult to do so when people have a vested interest in big rockets and porkus quo. To most of the rest of us, it was obvious quite some time ago that it would have been faster, cheaper, and safer to develop depots than monster rockets with inherent low flight rate.

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John, Vlad, Tom,

Glad you guys liked this. I wish we could’ve gotten to doing this analysis sooner after the initial concerns were raised by the JSC folks–to be fair I think they were probably arguing in good faith, LEO depots don’t work well with traditional departure techniques for interplanetary missions. Fortunately we found a good solution to the problem that even provides some useful benefits (like having a week or more to check out your interplanetary mission vehicle before committing to a long-duration trip, the ability to place roving depots into position for really high energy missions, etc).

I worry that my timing on these papers may have left them overshadowed a bit by BFR speculations. But I’m glad to have the writeups done. They’ve been on my plate for weeks.

~Jon

Man, this is all very very very reminiscent of Robert Farquar’s work…. (I don’t know how to spell the poor fellow’s name…)

Tim,

I’m sure my coauthors would be flattered to hear that!

~Jon