Cislunar Space · The Geometry, Physics and Arithmetic

What this is, and what it is not

The primer is “A Primer on Cislunar Space” by Holzinger, Chow and Garretson (AFRL, 3 May 2021, approved for public release). I read the PDF from the author’s Downloads folder and checked its key figures against the full text. This chapter rebuilds the geometry and the arithmetic behind those figures by hand and script. It does not give radiation doses (the primer does not either, and earlier lunar dose claims on this site were cut), does not design a mission, and does not say anything about where any agency will fly. Constants not in the primer (the Moon–Earth mass ratio, the sidereal month, the year, the speed of light, Earth’s gravitational parameter) are standard values written from memory and are labeled so.

§1 · Scale

The primer measures distance in geosynchronous radii: 1 GEO = 42,164 km from Earth’s centre. The mean Earth–Moon distance is 384,402 km, which is 9.12 GEO (the primer says 9.12). Light covers that in 1.282 s. A volume that reaches out to 12 GEO holds 12³ = 1,728 times the volume inside 1 GEO, because the volume of a sphere grows as the cube of its radius; the primer states that figure. It also says some transfers go beyond 1.5 million km, which is 35.6 GEO; the primer rounds that to 35.

Reading the primer’s Figure 1

Figure 1 in the primer (page 4, “Spatial scale in cislunar space”) lays a single axis out from Earth in GEO radii, numbered 1 to 13, with the geostationary orbit drawn around Earth, the Moon near 9, L1 just inside it and L2 just beyond. L4 and L5 are drawn off the axis. I do not reproduce the figure; here is what the model in §3 and §4 gives on the same ruler, measured from Earth’s centre as the primer does:

Object	Distance from Earth (GEO radii)	Figure 1 shows	Status
L1	7.74	between 7 and 8	computed; consistent
Moon	9.12	near 9	quoted (9.12), consistent
L2	10.65	between 10 and 11	computed; consistent
L3	9.05 (far side of Earth)	not drawn	computed
L4, L5	9.12 (same as the Moon, 60° off the axis)	marked off the axis	exact geometry

The reading I take from it: the whole near-Moon region the primer cares about, L1 to L2, spans only about 2.9 GEO radii, and GEO itself is the first unit of a ruler 13 units long. The positions I read off the figure are by eye, so the match is to the nearest whole unit, no better. figure read by eye

§2 · Two clocks: sidereal and synodic

The Moon circles Earth once in 27.32 days against the stars (sidereal). Earth goes round the Sun once in 365.256 days. The Moon–Sun angle repeats on the synodic month, found from the difference of the two angular rates:

1 / T_syn = 1 / T_sid − 1 / T_yr ⇒ T_syn = 29.53 days

That agrees with the primer’s 29.5 days. It matters because lighting on an orbit that repeats once per synodic month can be arranged to repeat too, which is why the primer singles out 1:1 synodic-resonant orbits for sensing. Earth turns 15.04° per hour; the Moon moves 13.18° per day. A ground station therefore cannot follow a cislunar object continuously: it hands the object off from site to site, which is the primer’s point about asynchronous tracking.

§3 · The model and its assumptions

The standard model for motion near the Earth–Moon system is the circular restricted three-body problem (CR3BP). It is a model, so each assumption is listed.

Earth and Moon are point masses on circular orbits about their common centre (the real Moon’s orbit has an eccentricity near 0.055; from memory).
The spacecraft has negligible mass and does not disturb them.
Only gravity acts. No Sun, no solar radiation pressure, no thrust, no non-spherical Moon. The primer says all of these matter, and that lunar gravity is highly uneven.
Units: distance in Earth–Moon separations, time so that the rotation rate is 1 (one unit = 4.3484 days), and total mass 1. With the mass ratio Moon/Earth = 1/81.3006 (from memory), μ = m_Moon / (m_Earth + m_Moon) = 0.0121506.
The frame rotates with the Moon. Earth sits at x = −μ, the Moon at x = 1 − μ.

Ω(x,y) = (x² + y²)/2 + (1−μ)/r₁ + μ/r₂ + μ(1−μ)/2
ẍ − 2ẏ = ∂Ω/∂x, ÿ + 2ẋ = ∂Ω/∂y

Here r₁ and r₂ are the distances to Earth and Moon. The constant term μ(1−μ)/2 only fixes the zero of the Jacobi constant used below; with it, the constant at L4 and L5 equals exactly 3.

Reading the primer’s Figure 2

Figure 2 in the primer (page 6, “Geometric and coordinate representation for the CR3BP”) is the top view of the model in §3: an inertial frame (X, Y) fixed to the stars, a rotating frame (x, y) turning with the Moon so that Earth and Moon stay on the x axis, dotted rings at 1, 5 and 10 GEO radii, and L1 to L5. The z axis points out of the page. The redraw below puts every point where the computation puts it, so the picture is to scale in a way a sketch is not: L4 and L5 are 9.12 GEO radii from Earth, 60° either side of the Moon. The 28° turn between the frames is arbitrary, chosen only to show that the two frames differ. redrawn, computed positions

§4 · The geometry of the five equilibrium points

An equilibrium is a point where a spacecraft at rest in the rotating frame stays at rest, so the gradient of Ω is zero. On the axis, y = 0, that is the single equation

x − (1−μ)(x+μ)/|x+μ|³ − μ(x−1+μ)/|x−1+μ|³ = 0

It has three real roots, one between the bodies and one beyond each. I solved it by bisection. The two off-axis points need no numerics: setting r₁ = r₂ = 1 satisfies both components, so L4 and L5 sit at the apexes of equilateral triangles built on the Earth–Moon line. That is a geometric fact, and the dotted triangles in the figure show it.

Point	x (units)	Where	Status
L1	0.8369	58,019 km from the Moon, toward Earth	computed
L2	1.1557	64,515 km beyond the Moon	computed
L3	-1.0051	381,677 km from Earth, opposite the Moon	computed
L4, L5	0.4879, ±0.8660	60° ahead of and behind the Moon on its orbit	computed (exact geometry)

Stability is a separate question. Linearizing near L4 and L5, motion is bounded only if μ < (1 − √(23/27))/2 = 0.0385 (Routh’s condition, a standard result from memory). For Earth–Moon, μ = 0.0121506 passes by a wide margin, which is why the primer calls L4 and L5 the only regions where objects stay a long time. L1, L2 and L3 are always unstable: a small error grows exponentially. This is linear theory in the idealised model. The primer notes that solar radiation pressure disturbs dust clouds near L4 and L5, so the real picture is messier.

Reading the primer’s Figure 3

Figure 3 in the primer (page 8, “2D projection of 3D spherical Zones in the CR3BP”) divides cislunar space by which bodies dominate the motion: a region around Earth, an Earth–Moon region, a region around the Moon, a Sun–Earth–Moon region, and outside that the Sun–Earth region. The primer says the partition is for discussion and intuition, loosely related to the Jacobi constant of §5, and that real propagation needs a full model. It gives no radii in its text, so the radii in my redraw are read off the figure by eye (about 5.7, 3.3 and 13.8 GEO radii) and are marked as such. Three things follow from the computed points. L1 and L2 (7.74 and 10.65 GEO radii) fall inside the Moon zone, as the figure shows. L3 (9.05 GEO radii on the far side) and L4 and L5 (9.12 GEO radii, both 60° off the axis) lie between the Earth–Moon circle and the outer circle, so all three sit in the Sun–Earth–Moon zone. Last, do not confuse the primer’s Moon zone (about 3.3 GEO radii by eye) with the Hill radius of §5, which is 1.45 GEO radii: they are different constructions and the numbers differ by more than a factor of two. The figure’s own definition of the boundaries is in the reference the primer cites, which I have not read. figure read by eye; boundaries not verified

§5 · The Jacobi constant and the gates at L1 and L2

In the CR3BP one quantity is conserved along every path: C = 2Ω(x,y) − (ẋ² + ẏ²). Since speed squared cannot be negative, a path with constant C can only visit places where 2Ω ≥ C. The curve 2Ω = C is a zero-velocity curve. Its shape depends on C, and the critical values are those at the equilibrium points:

Point	C (this convention)	Meaning	Status
L1	3.200	for C below this, the region around Earth connects to the region around the Moon through a neck at L1	computed
L2	3.184	below this, the neck at L2 opens to the outside	computed
L3	3.024	below this, the far side opens too	computed
L4, L5	3.000	the minimum of 2Ω is here; below 3 nothing is forbidden	computed

So “low-energy” routes in the primer’s sense are those where C sits just under C(L1) or C(L2): the gates are barely open, and a spacecraft needs very little extra energy to pass. The L1 and L2 points lie close to the edge of the Moon’s zone of dominance. A rough size for that zone is the Hill radius, a(μ/3)^1/3 = 61,274 km for a = 384,402 km. The match with L1 and L2 (58,019 and 64,515 km) is approximate, not exact.

§6 · Periods near L1 and L2, and a check against the primer

Expanding the equations to first order about L1 or L2 gives, for in-plane motion, one real pair of exponents (the instability) and one imaginary pair (oscillation). With c = μ/r₂³ + (1−μ)/r₁³ evaluated at the point, the oscillation frequency solves ω⁴ − (2−c)ω² + (1 + c − 2c²) = 0 and the growth rate λ solves λ² = [(c−2) + √(9c²−8c)]/2. Converted to days:

Point	Small-orbit period	Error grows by e in	Primer, larger orbits
L1 Lyapunov	11.7 days	1.5 days	12.2 days (quoted, checked)
L2 Lyapunov	14.7 days	2.0 days	14.5 days (quoted, checked)

The two columns are not meant to be equal. Orbits of larger amplitude have longer periods than the linear estimate, and the primer’s numbers come from one example transfer, so what the comparison shows is only that the linear model puts the periods at the right scale (about two weeks) and in the right order (L2 longer than L1). The primer’s example transfer from the 12.2-day L1 orbit to the 14.5-day L2 orbit takes 20.35 days; I did not recompute it. The growth times say why station-keeping is needed: a small error grows by a factor of e in about 1.5 to 2.0 days, so in linear theory an uncorrected error is multiplied by about 2937192 in 30 days (e^30/2.0, using the slower L2 rate). The linear estimate stops being valid long before that, but it makes the point: the error does not stay small. computed, linear theory

Try it: a path near L1 or L2

The page integrates the model above with a fourth-order Runge–Kutta step and plots the path in the rotating frame. It starts at rest a small distance from the chosen point. Because the point is unstable the path leaves; the Jacobi constant shown should stay near its starting value, which is a check on the integrator, not on reality.

Point Offset from the point (km) 100 Duration (days) 60

Not real: no Sun, no thrust, no lunar gravity field, circular lunar orbit. Treat it as a picture of the model.

§7 · Seeing across the distance

The primer says objects at lunar distance are fainter and that active sensing needs prohibitive power. Both statements follow from inverse-square laws. Take an object of the same size and reflectivity, seen at the same lighting phase. Reflected sunlight reaching an observer falls as 1/R², so moving from GEO to lunar distance (a factor of 9.12 in range from Earth’s centre) dims the object by (9.12)² = 83, which is 4.8 astronomical magnitudes. For a radar or laser that must send light out and catch the echo, the received power falls as 1/R⁴: (9.12)⁴ = 6,908 times weaker. The model ignores atmosphere, target shape, phase angle, antenna gain and sensor noise, and measures range from Earth’s centre where a real station is a few thousand kilometres away. It shows the scaling, not a detection limit.

§8 · Reaching and keeping an object there

A rough cost of reaching lunar distance uses the two-body “vis-viva” energy equation. From a 300 km circular parking orbit (radius 6,678.137 km, with μ_Earth = 398,600.4418 km³/s²) to an ellipse whose far point is at lunar distance, the burn is 3.11 km/s and the coast takes 4.98 days. This ignores the Moon’s gravity, the orbital plane, finite burn time and the capture burn, so it is an order-of-magnitude figure (computed, two-body). The primer says direct transfers take several days, which is consistent.

The primer says maintenance maneuvers are often in the mm/s range. A velocity error of 1 mm/s drifts an object 86.4 m in a day and 2.6 km in 30 days if nothing corrects it (linear drift only; the instability in §6 makes real growth faster). That is why telling a deliberate maneuver from a natural perturbation is hard: both are tiny against the distances involved.

§9 · What each statement is

Statement	Status
Distances, GEO ratio, 1,728, 1.5 million km, 29.5 d, 12.2 / 14.5 / 20.35 d, mm/s, Gateway orbit	quoted from the AFRL primer; checked against the PDF
Moon/Earth mass ratio, sidereal month, year, c, Earth GM, lunar eccentricity	standard values from memory; not read this session
L1, L2, L3 positions; Jacobi values; Hill radius; periods and growth times; synodic month; scaling factors; TLI burn	computed here (script)
L4 and L5 equilateral geometry	exact
Routh limit	standard result, from memory; value computed
Anything about radiation dose, shielding, habitats, power, or a particular mission	not established here

Sources

Holzinger, Chow, Garretson, A Primer on Cislunar Space, AFRL, 2021. Companion script: book6/cislunar-space-verify.py. Related: WP71 · WP136 · Hex Habitat Mars.