We’ve spoken recently about civilizations expanding throughout the galaxy in a matter of hundreds of thousands of years, a thought that led Frank Tipler to doubt the existence of extraterrestrials, given the lack of evidence of such expansion. But let’s turn the issue around. What would the very beginning of our own interstellar exploration look like, if we reach the point where probes are feasible and economically viable? This is the question Johannes Lebert examines today. Johannes obtained his Master’s degree in Aerospace at the Technische Universität München (TUM) this summer. He likewise did his Bachelor’s in Mechanical Engineering at TUM and was visiting student in the field of Aerospace Engineering at the Universitat Politècnica de València (UPV), Spain. He has worked at Starburst Aerospace (a global aerospace & defense startup accelerator and strategic advisory company) and AMDC GmbH (a consultancy with focus on defense located in Munich). Today’s essay is based upon his Master thesis “Optimal Strategies for Exploring Nearby-Stars,” which was supervised by Martin Dziura (Institute of Astronautics, TUM) and Andreas Hein (Initiative for Interstellar Studies).
by Johannes Lebert
Last year, when everything was shut down and people were advised to stay at home instead of going out or traveling, I ignored those recommendations by dedicating my master thesis to the topic of interstellar travel. More precisely, I tried to derive optimal strategies for exploring near-by stars. As a very early-stage researcher I was really honored when Paul asked me to contribute to Centauri Dreams and want to thank him for this opportunity to share my thoughts on planning interstellar exploration from a strategic perspective.
Figure 1: Me, last year (symbolic image). Credit: hippopx.com).
As you are an experienced and interested reader of Centauri Dreams, I think it is not necessary to make you aware of the challenges and fascination of interstellar travel and exploration. I am sure you’ve already heard a lot about interstellar probe concepts, from gram-scale nanoprobes such as Breakthrough Starshot to huge spaceships like Project Icarus. Probably you are also familiar with suitable propulsion technologies, be it solar sails or fusion-based engines. I guess, you could also name at least a handful of promising exploration targets off the cuff, perhaps with focus on star systems that are known to host exoplanets. But have you ever thought of ways to bring everything together by finding optimal strategies for interstellar exploration? As a concrete example, what could be the advantages of deploying a fleet of small probes vs. launching only few probes with respect to the exploration targets? And, more fundamentally, what method can be used to find answers to this question?
In particular the last question has been the main driver for this article: Before starting with writing, I was wondering a lot what could be the most exciting result I could present to you and found that the methodology as such is the most valuable contribution on the way towards interstellar exploration: Once the idea is understood, you are equipped with all relevant tools to generate your own results and answer similar questions. That is why I decided to present you a summary of my work here, addressing more directly the original idea of Centauri Dreams (“Planning […] Interstellar Exploration”), instead of picking a single result.
Below you’ll find an overview of this article’s structure to give you an impression of what to expect. Of course, there is no time to go into detail for each step, but I hope it’s enough to make you familiar with the basic components and concepts.
Figure 2: Article content and chapters
I’ll start from scratch by defining interstellar exploration as an optimization problem (chapter 2). Then, we’ll set up a model of the solar neighborhood and specify probe and mission parameters (chapter 3), before selecting a suitable optimization algorithm (chapter 4). Finally, we apply the algorithm to our problem and analyze the results (more generally in chapter 5, with implications for planning interstellar exploration in chapter 6).
But let’s start from the real beginning.
2. Defining and Classifying the Problem of Interstellar Exploration
We’ll start by stating our goal: We want to explore stars. Actually, it is star systems, because typically we are more interested in the planets that are potentially hosted by a star instead of the star as such. From a more abstract perspective, we can look at the stars (or star systems) as a set of destinations that can be visited and explored. As we said before, in most cases we are interested in planets orbiting the target star, even more if they might be habitable. Hence, there are star systems which are more interesting to visit (e. g. those with a high probability of hosting habitable planets) and others, which are less attracting. Based on these considerations, we can assign each star system an “earnable profit” or “stellar score” from 0 to 1. The value 0 refers to the most boring star systems (though I am not sure if there are any boring star systems out there, so maybe it’s better to say “least fascinating”) and 1 to the most fascinating ones. The scoring can be adjusted depending on one’s preferences, of course, and extended by additional considerations and requirements. However, to keep it simple, let’s assume for now that each star system provides a score of 1, hence we don’t distinguish between different star systems. Having this in mind, we can draw a sketch of our problem as shown in Figure 3.
Figure 3: Solar system (orange dot) as starting point, possible star systems for exploration (destinations with score ) represented by blue dots
To earn the profit by visiting and exploring those destinations, we can deploy a fleet of space probes, which are launched simultaneously from Earth. However, as there are many stars to be explored and we can only launch a limited number of probes, one needs to decide which stars to include and which ones to skip – otherwise, mission timeframes will explode. This decision will be based on two criteria: Mission return and mission duration. The mission return is simply the sum of the stellar score of each visited star. As we assume a stellar score of 1 for each star, the mission return is equal to the number of stars that is visited by all our probes. The mission duration is the time needed to finish the exploration mission.
In case we deploy several probes, which carry out the exploration mission simultaneously, the mission is assumed to be finished when the last probe reaches the last star on its route – even if other probes have finished their route earlier. Hence, the mission duration is equal to the travel time of the probe with the longest trip. Note that the probes do not need to return to the solar system after finishing their route, as they are assumed to send the data gained during exploration immediately back to Earth.
Based on these considerations we can classify our problem as a bi-objective multi-vehicle open routing problem with profits. Admittedly quite a cumbersome term, but it contains all relevant information:
- Bi-objective: There are two objectives, mission return and mission duration. Note that we want to maximize the return while keeping the duration minimal. Hence, from intuition we can expect that both objectives are competing: The more time, the more stars can be visited.
- Multi-vehicle: Not only one, but several probes are used for simultaneous exploration.
- Open: Probes are free to choose where to end their route and are not forced to return back to Earth after finishing their exploration mission.
- Routing problem with profits: We consider the stars as a set of destinations with each providing a certain score si. From this set, we need to select several subsets, which are arranged as routes and assigned to different probes (see Figure 4).
Figure 4: Problem illustration: Identify subsets of possible destinations si, find the best sequences and assign them to probes
Even though it appears a bit stiff, the classification of our problem is very useful to identify suitable solution methods: Before, we were talking about the problem of optimizing interstellar exploration, which is quite unknown territory with limited research. Now, thanks to our abstraction, we are facing a so-called Routing Problem, which is a well-known optimization problem class, with several applications across various fields and therefore being exhaustively investigated. As a result, we now have access to a large pool of established algorithms, which have already been tested successfully against these kinds of problems or other very similar or related problems such as the Traveling Salesman Problem (probably the most popular one) or the Team Orienteering Problem (subclass of the Routing Problem).
3. Model of the Solar Neighborhood and Assumptions on Probe & Mission Architecture
Obviously, we’ll also need some kind of galactic model of our region of interest, which provides us with the relevant star characteristics and, most importantly, the star positions. There are plenty of star catalogues with different focus and historical background (e.g. Hipparcos, Tycho, RECONS). One of the latest, still ongoing surveys is the Gaia Mission, whose observations are incorporated in the Gaia Archive, which is currently considered to be the most complete and accurate star database.
However, the Gaia Archive – more precisely the Gaia Data Release 2 (DR2), which will be used here* (accessible online  together with Gaia based distance estimations by Bailer-Jones et al. ) – provides only raw observation data, which include some reported spurious results. For instance, it lists more than 50 stars closer than Proxima Centauri, which would be quite a surprise to all the astronomers out there.
*1. Note that there is already an updated Data Release (Gaia DR3), which was not available yet at the time of the thesis.
Hence, a filtering is required to obtain a clean data set. The filtering procedure applied here, which consists of several steps, is illustrated in Figure 5 and follows the suggestions from Lindegren et al. . For instance, data entries are eliminated based on parallax errors and uncertainties in BP and RP fluxes. The resulting model (after filtering) includes 10,000 stars and represents a spherical domain with a radius of roughly 110 light years around the solar system.
Figure 5: Setting up the star model based on Gaia DR2 and filtering (animated figure from )
To reduce the complexity of the model, we assume all stars to maintain fixed positions – which is of course not true (see Figure 5 upper right) but can be shown to be a valid simplification for our purposes, and we limit the mission time frames to 7,000 years. 7,000 years? Yes, unfortunately, the enormous stellar distances, which are probably the biggest challenge we encounter when planning interstellar travel, result in very high travel times – even if we are optimistic concerning the travel speed of our probes, which are defined by the following.
We’ll use a rather simplistic probe model based on literature suggestions, which has the advantage that the results are valid across a large range of probe concepts. We assume the probes to travel along straight-line trajectories (in line with Fantino & Casotto  at an average velocity of 10 % of the speed of light (in line with Bjørk . They are not capable of self-replicating; hence, the probe number remains constant during a mission. Furthermore, the probes are restricted to performing flybys instead of rendezvous, which limits the scientific return of the mission but is still good enough to detect planets (as reported by Crawford . Hence, the considered mission can be interpreted as a reconnaissance or scouting mission, which serves to identify suitable targets for a follow-up mission, which then will include rendezvous and deorbiting for further, more sophisticated exploration.
Disclaimer: I am well aware of the weaknesses of the probe and mission model, which does not allow for more advanced mission design (e. g. slingshot maneuvers) and assumes a very long-term operability of the probes, just to name two of them. However, to keep the model and results comprehensive, I tried to derive the minimum set of parameters which is required to describe interstellar exploration as an optimization problem. Any extensions of the model, such as a probe failure probability or deorbiting maneuvers (which could increase the scientific return tremendously), are left to further research.
4. Optimization Method
Having modeled the solar neighborhood and defined an admittedly rather simplistic probe and mission model, we finally need to select a suitable algorithm for solving our problem, or, in other words, to suggest “good” exploration missions (good means optimal with respect to both our objectives). In fact, the algorithm has the sole task of assigning each probe the best star sequences (so-called decision variables). But which algorithm could be a good choice?
Optimization or, more generally, operations research is a huge research field which has spawned countless more or less sophisticated solution approaches and algorithms over the years. However, there is no optimization method (not yet) which works perfectly for all problems (“no free lunch theorem”) – which is probably the main reason why there are so many different algorithms out there. To navigate through this jungle, it helps to recall our problem class and focus on the algorithms which are used to solve equal or similar problems. Starting from there, we can further exclude some methods a priori by means of a first analysis of our problem structure: Considering n stars, there are η! possibilities to arrange them into one route, which can be quite a lot (just to give you a number: for n=50 we obtain 50!≈ 1064 possibilities).
Given that our model contains up to 10,000 stars, we cannot simply try out each possibility and take the best one (so called enumeration method). Instead, we need to find another approach, which is more suitable for those kinds of problems with a very large search space, as an operations researcher would say. Maybe you already have heard about (meta-)heuristics, which allow for more time-efficient solving but do not guarantee to find the true optimum. Even if you’ve never heard about them, I am sure that you know at least one representative of a metaheuristic-based solution, as it is sitting in front of your screen right now as you are reading this article… Indeed, each of us is the result of a thousands of years lasting, still ongoing optimization procedure called evolution. Wouldn’t it be cool if we could adopt the mechanisms that brought us here to do the next, big step in mankind and find ways to leave the solar system and explore unknown star systems?
Those kinds of algorithms, which try to imitate the process of natural evolution, are referred to as Genetic Algorithms. Maybe you remember the biology classes at school, where you learned about chromosomes, genes and how they are shared between parents and their children. We’ll use the same concept and also the wording here, which is why we need to encode our optimization problem (illustrated in Figure 6): One single chromosome will represent one exploration mission and as such one possible solution for our optimization problem. The genes of the chromosome are equivalent to the probes. And the gene sequences embody the star sequences, which in turn define the travel routes of each probe.
If we are talking about a set of chromosomes, we will use the term “population”, therefore sometimes one chromosome is referred to as individual. Furthermore, as the population will evolve over the time, we will speak about different generations (just like for us humans).
Figure 6. Genetic encoding of the problem: Chromosomes embody exploration missions; genes represent probes and gene sequences are equivalent to star sequences.
The algorithm as such is pretty much straightforward, the basic working principle of the Genetic Algorithm is illustrated below (Figure 7). Starting from a randomly created initial population, we enter an evolution loop, which stops either when a maximum number of generations is reached (one loop represents one generation) or if the population stops evolving and keeps stable (convergence is reached).
Figure 7: High level working procedure of the Genetic Algorithm
I don’t want to go into too much detail on the procedure – interested readers are encouraged to go through my thesis  and look for the corresponding chapter or see relevant papers (particularly Bederina and Hifi , from where I took most of the algorithm concept). To summarize the idea: Just like in real life, chromosomes are grouped into pairs (parents) and create children (representing new exploration missions) by sharing their best genes (which are routes in our case). For higher variety, a mutation procedure is applied to a few children, such as a partial swap of different route segments. Finally, the worst chromosomes are eliminated (evolve population = “survival of the fittest”) to keep the population size constant.
Side note: Currently, we have the chance to observe this optimization procedure when looking at the Coronavirus. It started almost two years ago with the alpha version; right now the population is dominated by the delta version, with omicron an emerging variant. From the virus perspective, it has improved over time through replication and mutation, which is supported by large populations (i.e., a high number of cases).
Note that the genetic algorithm is extended by a so-called local search, which comprises a set of methods to improve routes locally (e. g. by inverting segments or swapping two random stars within one route). That is why this method is referred to as Hybrid Genetic Algorithm.
Now let’s see how the algorithm is operating when applied to our problem. In the animated figure below, we can observe the ongoing optimization procedure. Each individual is evaluated “live” with respect to our objectives (mission return and duration). The result is plotted in a chart, where one dot refers to one individual and thus represents one possible exploration mission. The color indicates the corresponding generation.
Figure 8: Animation of the ongoing optimization procedure: Each individual (represented by a dot) is evaluated with respect to the objectives, one color indicates one generation
As shown in this animated figure, the algorithm seems to work properly: With increasing generations, it tries to generate better solutions, as it optimizes towards higher mission return and lower mission duration (towards the upper left in the Figure 8). Solutions from the earlier generation with poor quality are subsequently replaced by better individuals.
5. Optimization Results
As a result of the optimization, we obtain a set of solutions (representing the surviving individuals from the final generation), which build a curve when evaluated with respect to our twin objectives of mission duration and return (see Figure 9). Obviously, we’ll get different curves when we change the probe number m between two optimization runs. In total, 9 optimization runs are performed; after each run the probe number is doubled, starting with m=2. As already in the animated Figure 8, one dot represents one chromosome and thus one possible exploration mission (one mission is illustrated as an example).
Figure 9: Resulting solutions for different probe numbers and mission example represented by one dot
Already from this plot, we can make some first observations: The mission return (which we assume equal to the number of explored stars, just as a reminder) increases with mission duration. More precisely, there appears to be an approximately linear incline of star number with time, at least in most instances. This means that when doubling the mission duration, we can expect more or less twice the mission return. An exception to this behavior is the 512 probes curve, which flattens when reaching > 8,000 stars due to the model limits: In this region, only few unexplored stars are left which may require unfavorable transfers.
Furthermore, we see that for a given mission duration the number of explored stars can be increased by launching more probes, which is not surprising. We will elaborate a bit more on the impact of the probe number and on how it is linked with the mission return in a minute.
For now, let’s keep this in our mind and take a closer look at the missions suggested by the algorithm. In the figure below (Figure 10), routes for two missions with different probe number m but similar mission return J1 (nearly 300 explored stars) are visualized (x, y, z-axes dimensions in light years). One color indicates one route that is assigned to one probe.
Figure 10: Visualization of two selected exploration missions with similar mission return J1 but different probe number m – left: 256 available probes, right: 4 available probes (J2 is the mission duration in years)
Even though the mission return is similar, the route structures are very different: The higher probe number mission (left in Figure 10) is built mainly from very dense single-target routes and thus focuses more on the immediate solar neighborhood. The mission with only 4 probes (right in Figure 10), contrarily, contains more distant stars, as it consists of comparatively long, chain-like routes with several targets included. This is quite intuitive: While for the right case (few probes available) mission return is added by “hopping” from star to star, in the left case (many probes available) simply another probe is launched from Earth. Needless to say, the overall mission duration J2 is significantly higher when we launch only 4 probes (> 6000 years compared to 500 years).
Now let’s look a bit closer at the corresponding transfers. As before, we’ll pick two solutions with different probe number (4 and 64 probes) and similar mission return (about 230 explored stars). But now, we’ll analyze the individual transfer distances along the routes instead of simply visualizing the routes. This is done by means of a histogram (shown in Figure 11), where simply the number of transfers with a certain distance is counted.
Figure 11: Histogram with transfer distances for two different solution – orange bars belong to a solution with 4 probes, blue bars to a solution with 64 probes; both provide a mission return of roughly 230 explored stars.
The orange bars belong to a solution with 4 probes, the blue ones to a solution with 64 probes. To give an example on how to read the histogram: We can say that the solution with 4 probes includes 27 transfers with a distance of 9 light years, while the solution with 64 probes contains only 8 transfers of this distance. What we should take from this figure is that with higher probe numbers apparently more distant transfers are required to provide the same mission return.
Based on this result we can now concretize earlier observations regarding the probe number impact: From Figure 9 we already found that the mission return increases with probe number, without being more specific. Now, we discovered that the efficiency of the exploration mission w. r. t. routing decreases with increasing probe number, as there are more distant transfers required. We can even quantify this effect: After doing some further analysis on the result curve and a bit of math, we’ll find that the mission return J1 scales with probe number m according to ~m0.6 (at least in most instances). By incorporating the observations on linearity between mission return and duration (J2), we obtain the following relation: J1 ~ J2m0.6.
As J1 grows only with m0.6 (remember that m1 indicates linear growth), the mission return for a given mission duration does not simply double when we launch twice as many probes. Instead, it’s less; moreover, it depends on the current probe number – in fact, the contribution of additional probes to the overall mission return diminishes with increasing probe numbers.
This phenomenon is similar to the concept of diminishing returns in economics, which denotes the effect that an increase of the input yields progressively lower or even reduced increase in output. How does that fit with earlier observations, e. g. on route structure? Apparently, we are running into some kind of a crowding effect, when we launch many probes from the same spot (namely our solar system): Long initial transfers are required to assign each probe an unexplored star. Obviously, this effect intensifies with each additional probe being launched.
6. Conclusions and Implications for Planning Interstellar Exploration
What can we take from all this effort and the results of the optimization? First, let’s recap the methodology and tools which we developed for planning interstellar exploration (see Figure 12).
Figure 12: Methodology – main steps
Beside the methodology, which of course can be extended and adapted, we can give some recommendations for interstellar mission design considerations, in particular regarding the probe number impact:
- High probe numbers are favorable when we want to explore many stars in the immediate solar neighborhood. As further advantage of high probe numbers, mostly single-target missions are performed, which allows the customization of each probe according to its target star (e. g. regarding scientific instrumentation).
- If the number of available probes is limited (e. g. due to high production costs), it is recommended to include more distant stars, as it enables a more efficient routing. The aspect of higher routing efficiency needs to be considered in particular when fuel costs are relevant (i. e. when fuel needs to be transported aboard). For other, remotely propelled concepts (such as laser driven probes, e. g. Breakthrough Starshot) this issue is less relevant, which is why those concepts could be deployed in larger numbers, allowing for shorter overall mission duration at the expense of more distant transfers.
- When planning to launch a high number of probes from Earth, however, one should be aware of crowding effects. This effect sets in already for few probes and intensifies with each additional probe. One option to encounter this issue and thus support a more efficient probe deployment could be swarm-based concepts, as indicated by the sketch in Figure 13.
The swarm-based concept includes a mother ship, which transports a fleet of smaller explorer probes to a more distant star. After arrival, the probes are released and start their actual exploration mission. As a result, the very dense, crowded route structures, which are obtained when many probes are launched from the same spot (see again Figure 10, left plot), are broken up.
Figure 13: Sketch illustrating the beneficial effect of swarm concepts for high probe numbers.
Obviously, the results and derived implications for interstellar exploration are not mind-blowing, as they are mostly in line with what one would expect. However, this in turn indicates that our methodology seems to work properly, which of course does not serve as a full verification but is at least a small hint. A more reliable verification result can be obtained by setting up a test problem with known optimum (which is not shown here, but was also done for this approach, showing that the algorithm’s results deviate about 10% compared to the ideal solution).
Given the very early-stage level of this work, there is still a lot of potential for further research and refinement of the simplistic models. Just to pick one example: As a next step, one could start to distinguish between different star systems by varying the reward of each star system si based on a stellar metric, where more information of the star is incorporated (such as spectral class, metallicity, data quality, …). In the end it’s up to oneself, which questions he or she wants to answer – there is more than enough inspiration up there in the night sky.
Figure 14: More people, now
Assuming that you are not only an interested reader of Centauri Dreams but also familiar with other popular literature on that topic, you maybe have heard about Clarke’s three laws. I would like to close this article by taking up his second one: The only way of discovering the limits of the possible is to venture a little way past them into the impossible. As said before, I hope that the introduced methodology can help to answer further questions concerning interstellar exploration from a strategic perspective. The more we know, the better we are capable of planning and imagining interstellar exploration, thus pushing gradually the limits of what is considered to be possible today.
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