Deep Space Strategy · solmaps · How solmaps works · Computation status

Technical documentation

Deep Space Strategy organises its work in five modules: Geography (where sites are and how they move), Resources (what they hold), Topology (the temporal transfer graph), Infrastructure (hubs, depots, stations) and Strategy (measures, actors, governance). Its engine, solmaps, models travel in the solar system as a temporal graph: twelve sites, 132 directed routes, and a cost for every route on every departure date from November 2026 to the end of 2100. This page defines the graph, derives every formula the code uses, lists the checks against real missions, and places the work in the literature it builds on.

Status: experimental research prototype. The models simplify the physics (impulsive burns, patched conics). Results support comparisons between routes, dates and hubs. Code: github.com/bhanneke/solmaps; build steps in How solmaps works.

1. The idea: a timetable for the solar system
2. The temporal transfer graph
3. The cost of one edge
4. Launch windows
5. Journeys and the time-expanded graph
6. Slingshot edges
7. Hubs, depots and the rocket equation
8. Strategy measures
9. Validation
10. Scientific grounding
11. Limits and roadmap
12. References

1. The idea: a timetable for the solar system

Every site in the solar system moves on its own orbit, so the propellant needed to fly between two sites depends on the departure date. Earth to Mars costs 5.7 km/s from a 300 km Earth parking orbit in the November 2026 window and up to 33 km/s between windows. A map with one cost per route hides what matters for strategy: when a route opens, how long it stays open, and what waiting costs.

A railway map shows the lines; a timetable says when each train runs. solmaps builds the timetable. Network science calls such an object a temporal network: a graph whose edges carry time labels, so that only time-respecting paths are valid journeys [Holme & Saramäki 2012], [Kempe et al. 2002]. Public-transport routing solves the same problem on Earth with time-expanded graphs [Pyrga et al. 2008]. solmaps fills the edge labels with astrodynamics.

2. The temporal transfer graph

solmaps works with

• \(V\), 12 nodes: Earth, Venus, Mercury, Mars, Jupiter, Saturn, Ceres, Vesta, Psyche, Eros, Bennu and Ryugu. A planet node stands for a parking orbit around it; a small-body node stands for the body itself.
• \(E = \{(i,j) \in V \times V : i \ne j\}\), 132 directed edges. Direction matters: windows and costs differ from Mars to Earth and from Earth to Mars.
• \(\mathcal{T} = \{t_0 + 10k : k = 0,\dots,2708\}\), departure dates every 10 days from \(t_0\) = 2026-11-01 (JD 2461345.5) to the end of 2100.
• \(c(i,j,t)\), the cheapest Δv (km/s) to leave \(i\) on date \(t\) and arrive at \(j\).
• \(\tau(i,j,t)\), the flight time (days) that achieves \(c(i,j,t)\). A ship that leaves \(i\) at \(t\) arrives at \(t + \tau(i,j,t)\).

Each edge therefore has two time series, cost and flight time, over 2,709 dates.

3. The cost of one edge

Positions

Planet positions and velocities come from the JPL DE440 ephemeris [Park et al. 2021], read as planet-system barycentres through ESA's pykep [pykep]; the short DE440s kernel covers 1849 to 2150. The resource maps use JPL's approximate Keplerian elements [Standish & Williams 1992], valid 1800 to 2050. For the 21 small bodies the precomputed network uses daily JPL Horizons state vectors from 2026 to 2205, which include the planets' perturbations; between two days solmaps interpolates with cubic Hermite polynomials. After 2149 the planets also come from Horizons (DE441 barycentres). The in-browser engine of the growth map takes small-body elements from the JPL Small-Body Database [JPL SBDB] and moves the bodies on two-body Kepler orbits: with mean motion \(n = \sqrt{\mu_\odot / a^3}\), the mean anomaly at epoch \(t\) is \(M(t) = M_0 + n\,(t - t_\mathrm{epoch})\), and Kepler's equation \(M = E - e \sin E\) gives the eccentric anomaly by Newton iteration.

The transfer arc: Lambert's problem

Given position \(\mathbf r_1\) of node \(i\) at time \(t\), position \(\mathbf r_2\) of node \(j\) at \(t+T\), and the Sun's gravitational parameter \(\mu_\odot\), Lambert's problem asks for the conic arc that connects them in time \(T\). Its solution gives the ship's velocity \(\mathbf v_1\) after departure and \(\mathbf v_2\) before arrival. solmaps takes the single-revolution, prograde solution (a universal-variable solver in the browser, Izzo's algorithm in pykep [Izzo 2015]). With the node velocities \(\mathbf V_i\), \(\mathbf V_j\), the hyperbolic excess speeds are

Burns at both ends

A ship in a circular parking orbit of radius \(r\) around a body with gravitational parameter \(\mu\) reaches excess speed \(v_\infty\) most cheaply with one burn at that radius (the Oberth effect). Energy conservation on the escape hyperbola gives

The same expression gives the capture burn at arrival. Parking orbits: 300 km above Earth, Venus and Mars, 200 km above Mercury, 50 km above Ceres, circular orbits at the distance of Callisto (Jupiter) and Titan (Saturn). At the small asteroids gravity is negligible, so the cost is the rendezvous speed \(v_\infty\) itself.

Best flight time

For each date, solmaps tries 100 flight times \(\mathcal F_{ij}\), spaced logarithmically from \(0.3\,T_H\) to \(2.5\,T_H\) (at least 10 days, at most 100 years), where \(T_H\) is the Hohmann transfer time between the mean orbital radii, refines the three cheapest local minima and keeps the cheapest:

The minimising \(T\) is \(\tau(i,j,t)\). The engine computes this for every day and for all 870 routes between 30 bodies, about 2.6 billion Lambert solutions; the maps use every tenth day and the 132 routes between the 12 sites.

4. Launch windows

The cost series of an edge repeats with the synodic period of its two nodes. Bodies with orbital periods \(P_i\), \(P_j\) return to the same relative geometry after

For Earth and Mars, \(S = 779.94\) days [NSSDC Mars fact sheet]. The DE440 network finds Earth-to-Mars window bottoms at 5.68 km/s (November 2026), 5.87 (November 2028), 6.30 (December 2030), 6.13 (April 2033) and 5.79 (June 2035); Mars's eccentric orbit makes good and poor windows alternate over about 15 years.

solmaps detects windows from the series itself. A date index \(k\) inside the horizon \([k_0, k_0+H]\) is a candidate when

with the horizon's ends counted when the cost rises away from them. Candidates less than 60 days apart merge into one window at their cheapest date, and windows more than 3 km/s above the edge's best window drop out. Longer horizons therefore list more windows: from Earth, 13 within 6 months, 20 within a year, 29 within 26 months.

5. Journeys and the time-expanded graph

A journey is a sequence of legs \((v_{k-1}, v_k)\) with departure dates \(t_k\) such that each leg leaves only after the previous one has arrived:

The gap between arrival and the next departure is waiting time in a parking orbit. Waiting uses no propellant. Three questions follow: the cheapest departure within a waiting horizon; the earliest arrival under a Δv budget \(B\); the cheapest journey with stops before a deadline.

All three become ordinary shortest-path problems on a static, time-expanded graph [Pyrga et al. 2008]. Each pair (node \(v\), date index \(k\)) becomes a vertex, joined by

• waiting arcs \((v,k) \to (v,k+1)\) with cost 0;
• travel arcs \((i,k) \to \big(j,\,k + \lceil \tau(i,j,t_k)/10 \rceil\big)\) with cost \(c(i,j,t_k)\).

The graph has 12 × 2,709 = 32,508 vertices and about 390,000 arcs. Every arc points forward in time, so the graph has no cycles, and one pass in date order computes the minimum Δv \(D(v,k)\) to reach every vertex:

The earliest arrival within budget \(B\) is the first \(k'\) with \(D(j,k') \le B\). The cost is linear in the number of arcs, so route queries need no heuristic solver. The pages answer the first question today; the full journey queries are the next build step.

6. Slingshot edges

A slingshot route enters the graph as one more edge from its origin to its target, with its own cost and flight time, and competes with the direct edge on every date. A ship passing a planet keeps its speed relative to the planet and only turns. For excess speed \(v_\infty\), closest approach \(r_p\) and planet parameter \(\mu\), the turn angle \(\delta\) satisfies

solmaps searches flyby sequences with the MGA-1DSM model [Vasile & De Pascale 2006], [Izzo et al. 2007], implemented in pykep: each leg may contain one deep-space manoeuvre. The decision vector is

with launch epoch \(t_0\), launch direction \((u,v)\) and speed \(V_\infty\), and for each leg the flight time \(T_k\), the fraction \(\eta_k\) of the leg flown before the manoeuvre, the flyby plane angle \(\beta_k\) and radius \(r_{p,k}\). The objective adds the parking-orbit departure burn (Section 3), the deep-space manoeuvres and the capture burn at the target. pygmo [Biscani & Izzo 2020] optimises it with self-adaptive differential evolution [Brest et al. 2006] on eight islands, two independent starts per sequence and two-year launch window, with leg flight times capped at 1.9 Hohmann times.

Today's sequences have up to two flybys of Venus, Earth, Mars or Jupiter, from low Earth and low Mars orbit, to Jupiter, Saturn, Mercury, Ceres, Vesta and Psyche. The table compares the best slingshot route from low Earth orbit with the direct route in the same two-year launch window, over the 36 windows from 2027 to 2098; the median covers the windows where the slingshot costs less.

From low Mars orbit the gains are smaller. The most frequent slingshot is a return to Mars itself: the ship leaves on an orbit that meets Mars again two Mars years later, and the flyby sends it on.

Drawn paths. The 3D atlas draws routes from low Mars orbit along the optimiser's own trajectory: pykep rebuilds each route from its stored decision vector, deep-space manoeuvres included, and the page samples it at 80 points plus the exact flyby moments. Every rebuilt route ends within 1,200 km of its target planet on the arrival date. Routes from Earth come from the first run, which stored only launch date, arrival date and sequence. For those, the page estimates the flyby dates: it splits the flight between launch and arrival at every candidate date, solves the Lambert legs, and keeps the split with the lowest score

whose last term penalises flybys that would need a burn. The page labels these paths as estimated.

7. Hubs, depots and the rocket equation

A ship with exhaust speed \(v_e\) needs a start mass of \(e^{\Delta v / v_e}\) per kilogram delivered (Tsiolkovsky). Flown directly from low Earth orbit to destination \(d\), all of it comes from Earth. Via a hub \(h\) that makes propellant from local water, only the leg to the hub uses Earth propellant, and the rocket equation restarts there. The share of Earth-launched mass the hub saves is

with \(\Delta v_\mathrm{in} = c(\text{Earth}, h, t)\) and \(\Delta v_\mathrm{direct} = c(\text{Earth}, d, t)\), each the cheapest within the waiting horizon. A destination counts only when the hub is cheaper to reach than the destination and the onward leg \(c(h,d,t)\) is cheaper than the direct one (detour rule); otherwise \(L_h = 0\). Per kilogram delivered, the hub supplies

kilograms of propellant from its surface, where \(\Delta v_\mathrm{lift}\) lifts it to the depot: 2.60 km/s from the lunar surface to Earth–Moon L1, 4.1 km/s from Mars's surface to low orbit, at most 0.52 km/s at Ceres, about zero at Bennu and Ryugu. The growth map and the station page take any specific impulse, with presets of 370 s (methane/oxygen, \(v_e\) = 3.6 km/s), 450 s (hydrogen/oxygen, 4.4 km/s) and 900 s (nuclear thermal, 8.8 km/s). With hydrogen/oxygen, an Earth–Moon L1 depot fed by lunar ice saves 52–84% of Earth-launched mass over 2026–2100 with slingshot routes included, and 63–84% with direct routes only; the onward leg leaves the hub only after the ship has arrived there.

Any engine. The transfer costs \(\Delta v\) come from planet positions and Lambert's problem alone, so one precomputed graph serves every high-thrust engine. The exhaust speed \(v_e = I_\mathrm{sp}\, g_0\) enters only through the rocket equation above. The station page stores the fuel saved per candidate orbit at 41 exhaust speeds from 2 to 200 km/s (an Isp of about 200 to 20,000 s) and interpolates linearly in \(\ln v_e\) for the Isp a reader enters; the largest interpolation error is 0.03 percentage points. Electric engines thrust for months and need more \(\Delta v\) than these impulsive transfers; the routing engine computes their continuous-thrust paths separately (docs/ENGINE.md).

Depot orbits. A depot in a solar orbit adds a node with its own time-varying edges. solmaps scored 771 coplanar candidate orbits on two costs, the average cheapest transfer from the depot to eight destinations over 20 years (60-day steps, up to 120 days of waiting) and the average supply cost from low Earth orbit, and reports the Pareto set.

8. Strategy measures

Whole network and go-return imbalance. For a date \(t\), \(T_{ab}\) is the earliest arrival at \(b\) for a ship leaving \(a\), waiting included, with the journey planner's settings. The 3D atlas places every node so that distances match the two-way average \((T_{ab} + T_{ba})/2\) as closely as possible: classical scaling for the first date, then minimisation of the stress

started from the previous date's layout so the picture moves smoothly. Any layout shows only the symmetric part of \(T\). The antisymmetric part appears as the pair imbalance \(I_{ab}\), from −1 to +1, and as the network imbalance \(A(t)\), its share of the whole. Averaged over every six months from 2026 to 2085, \(I_{ab}\) separates the two sources of imbalance. Launch windows recur, so the imbalance from their timing averages out. What remains is structural. It comes from the orbits themselves (eccentricity and inclination) and from the network (flybys or refuelling available at one end only). On 2026-11-01, with refuelling at stops and at most 8 km/s per leg, Earth to Eros takes 2.2 years and the return 4.2 years (\(I = -31\%\)), and the average over 2026–2085 is +1%: window timing. Earth and Jupiter differ by −10% on average: a lasting imbalance. \(A(t)\) ranges from 10% to 30% over the century.

9. Validation

The optimiser minimises Δv only. Real missions also respected launch energy, arrival dates and, for Dawn, ion propulsion, so solmaps often picks a slower and cheaper arrival.

10. Scientific grounding

solmaps joins four bodies of work that so far stand apart: trajectory design, space logistics, temporal networks and space strategy.

Trajectory design. Mission designers search launch windows and flyby sequences for one mission at a time. NASA's Trajectory Browser [NASA Ames] offers searchable precomputed trajectories to planets and small bodies. The flyby search in solmaps rests on the MGA-1DSM formulation and its global optimisation [Vasile & De Pascale 2006], [Izzo et al. 2007], on Izzo's Lambert solver [Izzo 2015] and, for pruning sequences, on the Tisserand graph [Strange & Longuski 2002]; Earth–Mars cyclers go back to Aldrin's work [Byrnes et al. 1993]. solmaps applies these methods to all site pairs and every date to 2100.

Space logistics. Ishimatsu et al. model the Earth–Moon–Mars logistics system as a time-expanded network with Δv-weighted arcs and multicommodity flows [Ishimatsu et al. 2016]. Theirs is the closest operations-research precedent for solmaps, built for one central planner. Elvis shows that the number of economically reachable ore-bearing asteroids depends strongly on the Δv cutoff [Elvis 2014], so transfer cost shapes resource geography.

Temporal networks. Network science defines temporal networks and time-respecting paths [Holme & Saramäki 2012], [Kempe et al. 2002]. Railway timetable research turns them into fast queries on time-expanded and time-dependent graphs [Pyrga et al. 2008]. solmaps applies both to orbital transfers.

Space strategy and governance. Dolman's astropolitics reads gravity wells, orbits and choke points as geopolitical terrain [Dolman 2002]. Klein applies Corbett's maritime strategy to space: command of space is relative and temporary, and lines of communication sit at its centre [Klein 2006]. Bowen ties orbital geography to war on Earth [Bowen 2020], Ziarnick treats space power as developed national capability [Ziarnick 2015], the National Defense University volume set out the theoretical agenda [Lutes & Hays 2011], and the US Space Force's capstone doctrine names space mobility and logistics [USSF 2020]. Deudney gives the main critique of space expansion as a long-run security problem [Deudney 2020]. Governance sets the rules on the map: the Outer Space Treaty bans national appropriation in Article II [Outer Space Treaty 1967], and the Artemis Accords cover space resources (Section 10) and safety zones (Section 11) and reach the Earth–Moon Lagrange points [Artemis Accords 2020]. Mendenhall argues that governance should follow from space's own physical features [Mendenhall 2018], and the AFRL primer describes cislunar geography for planners [Holzinger et al. 2021].

The gap. Strategy writers describe orbital terrain in words, trajectory tools compute it one mission at a time, and logistics models optimise for a single planner. Our literature search found no open dataset that resolves transfer costs between all major sites over time and computes strategic measures on them. solmaps aims to supply that layer.

11. Limits and roadmap

• The maps use impulsive burns and single-revolution conics: no low-thrust transfers (the engine computes them separately), multi-revolution arcs, aerocapture or lunar flybys yet. The next data release adds several revolutions and flights of up to 100 years on every route.
• The in-browser engine moves small bodies on two-body orbits without planetary perturbations, so its small-body costs lose accuracy over decades; the precomputed network uses the perturbed Horizons states.
• The 10-day date grid can miss the exact bottom of a narrow window by a few days. Depot candidates are coplanar.
• Roadmap: waiting horizons of 5 and 10 years; the network to 2150 (DE440s) and beyond (DE440 to 2650, DE441 for longer spans); up to three and four flybys with Saturn as a flyby planet; sequences enumerated and pruned with the Tisserand graph [Strange & Longuski 2002]; hub origins (Earth–Moon L1, Ceres, Vesta, Psyche, Jupiter); new targets and return legs; Earth–Mars cyclers [Byrnes et al. 1993]; the screening of every asteroid and comet in JPL's Small-Body Database.
• Idea: positioning human-made stations. Choosing the orbits (and so speed and phase) of \(k\) stations from a candidate set \(\mathcal O\) so that the network becomes cheapest to travel is a selection problem with pairwise interactions, since one station's value depends on which others exist. With \(x_o \in \{0,1\}\) for each candidate orbit it takes the QUBO form [Lucas 2014] $$\min_{\mathbf x \in \{0,1\}^{|\mathcal O|}} \ \sum_{o} a_o x_o + \sum_{o \lt o'} b_{oo'}\, x_o x_{o'} + \lambda\Big(\sum_o x_o - k\Big)^2,$$ where \(a_o\) is a station's stand-alone cost to the network, \(b_{oo'}\) the overlap or complementarity of two stations, and \(\lambda\) enforces \(k\) stations. Choosing 5 of 5,000 orbits allows about \(2.6 \times 10^{16}\) combinations. The plan is to solve it on quantum and hybrid solvers such as QAOA [Farhi et al. 2014] and against strong classical solvers, and to report where each wins. Build order, fleet scheduling and flyby-sequence choice share the structure. Route queries stay classical (Section 5).

12. References

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Deep Space Strategy is a research programme by Björn Hanneke; solmaps is its open engine. Code under the MIT licence, data and documentation under CC BY 4.0; source at github.com/bhanneke/solmaps.