D - Interplanetary Geometry & The Month-Gap Heuristic

A Low-Resolution Mental Map of a Moving Solar System

D.1 Purpose

Solarian's twelve-position astronomical calendar was designed to make local orbital calendars interoperable. It produces another useful property almost by accident.

Where two worlds use phase-valid Solarian astronomical calendars, their orbital-month positions provide rough information about their relative geometry around the Sun.

That geometry, in turn, provides rough information about:

• interplanetary separation;

• communications light time; and

• whether a particular pair of worlds is towards the favourable or unfavourable end of its normal range cycle.

This appendix explains that relationship.

It does not propose a navigation method. It does not replace an ephemeris. It does not claim that a calendar month can predict exact communications delay.

Its purpose is much simpler: Can two ordinary calendar dates give a person a useful first-order mental picture of where two worlds are relative to one another?

The answer is yes — within clear limits.

D.2 Why the common twelve-position grammar matters

Standard Solarian astronomical calendars share: one year-start direction

And: twelve corresponding orbital positions

Month 1 on Mars and Month 1 on Jupiter therefore refer, approximately, to the same directional part of their respective heliocentric orbits.

Month 7 lies roughly half an orbit around from Month 1. The worlds reach those month positions at very different speeds. That does not matter.

The important shared property is not: when they reach the position

But: what the position means

So: Janmar · Jancer · Janjov · Jansat

all indicate roughly corresponding portions of the heliocentric clock.

Likewise: Julmar · Julcer · Juljov · Julsat

indicate approximately the opposite side of that clock.

The suffix identifies the world. The month stem carries rough directional information.

D.3 Circular month gap

Suppose two worlds have orbital-month numbers: M₁

And: M₂ where each month is numbered from 1 to 12.

First take the ordinary difference: d = |M₁ − M₂|

Because the calendar is circular, Month 12 and Month 1 are neighbours rather than eleven positions apart.

So the circular month gap, G, is:

G = the smaller of d and (12 − d)

The result can only be: 0, 1, 2, 3, 4, 5 or 6

This is the entire heuristic.

D.4 Month-gap interpretation

Examples:

Month 12 versus Month 1 → gap 1

Month 11 versus Month 1 → gap 2

Month 1 versus Month 4 → gap 3

Month 1 versus Month 7 → gap 6

The maximum possible gap is therefore six.

D.5 Converting gap into a rough angle

The twelve-position orbital clock divides a complete revolution into twelve nominal sectors.

One position therefore represents approximately: 360° ÷ 12 = 30°

So the gap can be translated into a crude relative angle: Approximate angular separation = 30° × month gap

This is deliberately coarse.

The actual angular difference may be larger or smaller because real orbits are eccentric and Solarian months are equal-time civil approximations rather than exact 30° sectors.

D.6 The circular-orbit model

The geometry becomes easiest to understand if we temporarily imagine two worlds travelling on circular, coplanar orbits.

Let their characteristic orbital radii be: r₁

And: r₂ and their angular separation be: θ

Then the straight-line distance between them follows from ordinary triangle geometry:

distance² = r₁² + r₂² − 2 × r₁ × r₂ × cos(θ)

or, written as a direct estimate:

estimated distance = square root of
[r₁² + r₂² − 2r₁r₂ cos(θ)]

For the month-gap heuristic: θ ≈ 30° × G

This gives a very simple phase-aware range estimate requiring only:

• the identity of the two worlds;

• a characteristic orbital radius for each; and

• their current Solarian month positions.

That is almost exactly the level of information available from the calendar itself.

D.7 What happens at gap zero

At: G = 0 the approximate angle is: 0° and cos(0°) = 1.

The distance expression therefore reduces to: |r₂ − r₁|

In the circular model, the two worlds are in approximately the same heliocentric direction.

Their separation is therefore towards the difference between their orbital radii.

For that particular pair, this is broadly the favourable end of the distance cycle.

D.8 What happens at gap six

At: G = 6 the approximate angle is: 180° and cos(180°) = −1.

The distance therefore becomes approximately: r₁ + r₂

The worlds lie on broadly opposite sides of the Sun.

For that particular pair, this is towards the unfavourable end of the distance cycle.

So even before considering real orbital complexity, the basic reason the heuristic works is obvious:

same direction → difference of radii

opposite directions → sum of radii

The positions between those extremes produce intermediate separations.

D.9 Destination provides the scale

Month gap does not create one universal distance scale. That point is essential.

Earth and Mars at gap six may be far apart for Earth and Mars.

Earth and Neptune at gap zero remain vastly farther apart in absolute terms.

The month gap therefore answers: Where are we within this pair's normal geometry cycle?

It does not answer: How far apart are all gap-two worlds?

A person's useful mental model contains two pieces of information.

First: What broad distance class does this destination belong to?

Second: Where within that pair's favourable-to-unfavourable geometry cycle are we now?

So, in broad civil terms:

Earth–Mars is a minutes-scale relationship.

Earth–Jupiter is a tens-of-minutes relationship.

Earth–Neptune is an hours-scale relationship.

The month gap modifies the expectation within that scale.

D.10 From distance to light time

Once approximate distance is known, approximate one-way light time follows directly:

light time = distance ÷ speed of light

For human intuition, however, even this calculation is often unnecessary.

The useful mental shorthand is simply:

gap near 0 → better end of the pair's communications range

gap near 6 → worse end

with the middle gap values describing broad movement between those states.

The exact numerical light time remains a software and ephemeris problem.

D.11 Worked example: Earth and Ceres

Suppose Earth is around Month 1 and Ceres is in: Jancer also Month 1.

Then: G = 0

The calendar indicates broadly corresponding heliocentric directions.

Now leave Earth around Month 1 and move Ceres to: Julcer Month 7.

Then: G = 6

The two dates now indicate broadly opposite heliocentric directions.

Even without knowing either world's exact longitude, a reader can immediately infer that the second geometry is likely to produce a substantially greater separation than the first.

The dates have become a crude map.

D.12 Worked example: Mars and Jupiter

Suppose Mars is in: Marmar and Jupiter is in: Marjov

Both are Month 3. So: G = 0

They occupy broadly corresponding directions around the shared heliocentric clock.

Now compare: Marmar With: Sepjov Month 9. The gap is: 6

The worlds are now broadly opposite.

Again, no precise state vector is required to understand the first-order change in geometry.

D.13 Why month names help

The reference naming convention makes this relationship visible.

The first part of: Janmar And: Janjov carries the same twelve-position identity.

So do: Julcer And: Julsat

A person does not need to remember that these are technically Months 1 and 7.

The shared stems expose the relationship directly.

This is one reason the reference notation is useful even where local cultures later adopt entirely different spoken month names.

The underlying structural month number remains available to software and education.

D.14 The validation question

The month-gap heuristic sounds plausible on circular diagrams.

The important question is whether it survives contact with real orbital geometry.

The underlying Solarian analysis therefore tested a deliberately coarse estimator.

The estimator was not allowed access to precise instantaneous planetary positions.

Each world was represented only by:

• a characteristic orbital radius; and

• its current Solarian orbital-month position.

That information was compared with the actual interplanetary geometry derived from the orbital model.

A second comparison used a phase-blind alternative that knew the broad orbital scales but not the current Solarian month position.

The purpose of the exercise was not to discover the best possible range estimator.

A precise ephemeris would win that contest rather comprehensively.

The question was:

Does adding the Solarian month position contribute useful information beyond simply knowing which two worlds are involved?

D.15 What the analysis found

Across most of the tested Earth-relative and Mars-relative pairings: yes

Adding month position materially improved the coarse estimate compared with the phase-blind alternative.

The size of the improvement varied. That is expected.

Worlds differ in:

• orbital eccentricity;

• radial variation;

• orbital orientation;

• inclination;

• and how faithfully their particular Solarian month represents heliocentric phase.

The result therefore does not support the claim: month number accurately predicts distance

It supports the narrower claim: month number contains useful phase information that would otherwise be absent.

D.16 The stronger aggregate result

A second result is easier to interpret.

Across the tested Earth-reference and Mars-reference pairings, the mean actual one-way light time increased monotonically as the circular month gap increased:

0 → 1 → 2 → 3 → 4 → 5 → 6

That is important. It means the real orbital data preserved the broad ordering predicted by the simple clock-face model.

On average: corresponding month positions produced more favourable geometry;

And: opposite month positions produced less favourable geometry.

The individual measurements overlapped. The ordering of the means remained. The heuristic was coarse.

It was not meaningless.

D.17 Earth–Mars as the clearest example

Earth and Mars provide the easiest example to understand.

In the underlying analysis, configurations around: month gap 0

produced mean one-way light times of roughly: 5 minutes

Configurations around: month gap 6

produced mean one-way light times of roughly: 21 minutes

Those values should not be turned into another table people are expected to memorise.

Their purpose is to demonstrate scale. Knowing only the month gap shifts the expectation from one end of the Earth–Mars communications cycle towards the other.

A person told: Mars is around gap zero from Earth should expect broadly favourable geometry.

A person told: Mars is around gap six should expect broadly unfavourable geometry.

The exact answer may still differ substantially on a particular date.

The mental model has nevertheless done useful work.

D.18 Why the intermediate gaps overlap

It would be tempting to turn the seven gap values into seven precise distance bands.

That would be wrong. Real orbits are not circular.

A body may be:

• nearer perihelion;

• nearer aphelion;

• above or below another orbit's plane;

• displaced from the nominal month-sector centre; or

• on an orbit whose phase fidelity is less clean than another's.

So an actual configuration at: gap 2

can sometimes be farther apart than another configuration at: gap 3

Neighbouring categories overlap. What matters is the broad statistical and geometric ordering.

The heuristic is therefore closer to: near side / middle / far side than to a precision ruler.

D.19 Radial variation

The simple estimator assigns each world a characteristic orbital radius.

A real elliptical orbit does not possess one fixed radius. Its distance from the Sun changes continuously.

That means even if the angular estimate were perfect, the distance estimate would still contain error.

For low-eccentricity worlds this effect is relatively modest. For more eccentric bodies it can become substantial.

This is another reason the month-gap heuristic should never be confused with an operational range calculation.

D.20 Orbital inclination

The simple clock-face model is essentially two-dimensional.

Real orbits are three-dimensional.

Bodies can be inclined relative to the reference ecliptic and therefore separated vertically as well as within the projected heliocentric plane.

For most major planetary relationships the first-order planar model remains useful for civil intuition.

For significantly inclined small-body orbits, the approximation can degrade.

Again, exact geometry belongs to an ephemeris.

D.21 Month-position error

Appendix G establishes another source of uncertainty.

A Solarian orbital month is itself only an approximate indication of heliocentric phase.

If a calendar's Month 5 corresponds imperfectly to the nominal Month 5 sector, then the month-gap estimate inherits that error.

This leads to a strict dependency:

phase-valid calendar first

month-gap heuristic second

Specialised astronomical cases, including the illustrative Pluto calendar and Mercury’s independent chronology with a specialised phase proxy, cannot be inserted into the ordinary month-gap estimator as though all twelve civil month names carried equivalent directional meaning. A separately specified, evidence-validated phase display would be required first.

D.22 Independent calendars can still participate

Independent calendars are not excluded from the month-gap idea simply because their mears and orbital years run on separate clocks. Venus is a useful example: the independently calculated twelve-month astronomical sequence satisfied the phase criterion in the tested DE440 window.

Mercury provides the counterexample. Its independent twelve- or thirteen-week astronomical year remains chronologically meaningful, but its nominal twelve-month labels do not pass the standard phase-proxy test. Those labels therefore must not be used as an ordinary month-gap position estimate. Mercury could participate only through a separately specified specialised angular-phase display with its own validated mapping.

The requirement is phase fidelity, not an integer Repeat Factor. Calendar independence alone neither permits nor prevents a meaningful heliocentric directional heuristic.

D.23 Planetary systems share one position

A moon inheriting the astronomical calendar of its parent planetary system also inherits that system's heliocentric month position.

So, for this heuristic:

Titan uses Saturn's heliocentric position.

Ganymede uses Jupiter's.

Triton uses Neptune's.

The moon's local orbit around its parent planet is irrelevant to the first-order interplanetary geometry.

That local motion matters for transfers and local operations.

It does not materially change which side of the Sun the entire planetary system occupies.

D.24 The heuristic does not predict conjunction hazards

A low month gap does not necessarily mean unobstructed communications.

For example, two worlds can occupy broadly similar or opposing heliocentric directions while the Sun lies inconveniently close to the signal path as seen from one world.

Solar conjunction effects depend on actual geometry, angular separation from the Sun and operational radio conditions.

Month gap does not model those effects.

It therefore SHALL NOT be used to schedule critical communications around conjunction.

D.25 Nor does it predict travel time

Interplanetary travel does not follow a straight-line light path. A spacecraft trajectory depends on:

• propulsion system;

• departure and arrival velocities;

• gravitational environment;

• transfer geometry;

• mission constraints; and

• energy or acceleration strategy.

Two worlds being closer together does not automatically mean a particular spacecraft can travel between them proportionally faster.

The month-gap heuristic is therefore about: geometry and communications intuition

Not: mission duration

D.26 Precision remains one query away

In a mature Solar-System civilisation, precise positional information should be trivial to obtain.

That is not an argument against the heuristic. It is the reason to keep the heuristic modest.

People on Earth can obtain exact latitude and longitude instantly.

They still benefit from knowing that:

Paris is close to London.

Tokyo is far east of it.

New York is several time zones away.

The useful human model is not the most precise possible model.

It is the simplest model that preserves enough structure to guide expectation.

Solarian month position may provide that same function for moving interplanetary geography.

D.27 Why this matters more with many settlements

With five important destinations, people can simply memorise the current communication delay to each.

With five hundred, that approach becomes less attractive. With thousands, it becomes absurd.

A mature Solar-System civilisation may contain settlements on:

• major planets;

• moons;

• asteroids;

• independent habitats;

• mining sites;

• scientific stations; and

• transport or industrial nodes.

Exact information will remain abundant. Human attention will not. The common twelve-position grammar therefore acts as a compression system.

It gives every phase-valid settlement a rough place on the same mental clock.

D.28 Static geography versus moving geography

Terrestrial geographical relationships are mostly stable. Paris remains southeast of London.

Australia does not move to the opposite side of Earth during the year.

Solar-System geography is different.

The worlds continuously change their relative positions.

So a useful interplanetary mental map needs to encode not only: where a place belongs

But: where it is now

The astronomical calendar provides a surprisingly cheap way to carry part of that information.

DH.29 The twelve-pixel map

The deepest way to understand the heuristic is as a very low-resolution map.

Imagine the Sun in the centre. Place twelve positions around it.

Assign every phase-valid Solarian world to its current orbital month.

The result is, effectively: a twelve-pixel map of the Solar System

As an astronomical instrument, that is dreadful. As a mental model, it may be remarkably efficient.

From two dates alone, a person can infer whether two worlds are:

• broadly aligned;

• modestly separated;

• around a quarter orbit apart;

• or close to opposite sides of the heliocentric clock.

Combine that with basic knowledge of the destinations and it becomes possible to form a rough expectation of communications delay without consulting anything else.

D.30 A human-use version

For ordinary civil use, the full geometry can be reduced to three questions.

Question 1

What kind of destination is it? Mars-scale? Jupiter-scale? Neptune-scale? That tells you the broad communications magnitude.

Question 2

What is the circular month gap? 0 through 6. That tells you where the current geometry sits roughly within the pair's range cycle.

Question 3

Do I need the real number? If yes: look it up

The heuristic should make that final answer easier to anticipate.

It should never discourage obtaining it.

D.31 Appropriate uses

The month-gap heuristic may be useful for:

• ordinary geographical literacy;

• education;

• rough communications expectations;

• understanding whether a relationship is becoming more or less delayed;

• cross-world scheduling intuition;

• journalism and public explanation;

• strategic planning where precise timing is unnecessary; and

• mentally locating large numbers of settlements around the Solar System.

D.32 Inappropriate uses

It SHALL NOT be used as the sole basis for:

• spacecraft navigation;

• manoeuvre planning;

• precision ranging;

• critical communications scheduling;

• conjunction avoidance;

• mission departure windows;

• orbital operations;

• safety-critical timing;

• scientific ephemerides; or

• any decision requiring precise light time.

Those applications require actual state vectors and appropriate computational models.

D.33 Validation requirements for future use

A future formal implementation of the heuristic SHOULD validate it against the relevant settlement population.

For each tested world pair, the validation record SHOULD include:

1. the profile versions used;

2. the astronomical reference interval;

3. actual inter-world range from authoritative ephemerides;

4. actual one-way light time;

5. Solarian orbital-month positions;

6. circular month gap;

7. characteristic-radius phase-aware estimate;

8. a clearly defined phase-blind comparison;

9. absolute and relative estimation error;

10. mean actual OWLT by gap category;

11. overlap between neighbouring gap categories; and

12. any periods excluded because one calendar fails phase-fidelity requirements.

The purpose is not to certify a precision estimator. It is to establish whether the low-resolution month-gap ordering remains useful for the population being considered.

D.34 What has already been demonstrated

The Solarian analysis supporting this book establishes three useful findings.

Finding 1

Month position contains real geometric information. Across most tested Earth-relative and Mars-relative pairings, adding Solarian month position improved the coarse estimate over a phase-blind alternative.

Finding 2

The broad clock-face ordering survives real orbital geometry. Across the tested Earth- and Mars-reference pairings, mean actual one-way light time increased monotonically as month gap increased from: 0 to 6

Finding 3

The effect is large enough to matter intuitively.

For Earth–Mars, mean one-way light time moved from roughly: 5 minutes at gap 0

To: 21 minutes at gap 6

That is not a precision prediction. It is a materially different communications environment.

D.35 What has not been demonstrated

The analysis does not establish that:

• every individual observation follows the monotonic ordering;

• gap 3 always produces a longer delay than gap 2;

• one gap value corresponds to one universal distance;

• all Solar-System bodies perform equally well;

• specialised high-eccentricity calendars can use the same heuristic;

• the model is appropriate for navigation;

• month gap alone is sufficient to estimate exact one-way light time; or

• the heuristic should influence the underlying Solarian calendar design.

Those would be stronger claims than the evidence supports.

D.36 The core insight

The communications heuristic is not one of Solarian's foundational requirements.

The architecture would remain useful without it. That is precisely why the result is interesting.

A common astronomical grammar intended for calendar interoperability also produces a common geographical grammar.

The same twelve positions mean approximately the same direction around one Sun.

Once that exists: two dates become a crude angle

And: a crude angle becomes a rough distance intuition

And: rough distance becomes a communications intuition

No additional civil unit is required. No new notation is required. The information was already sitting inside the dates.

D.37 Summary

The Solarian month-gap heuristic works as follows:

1. Read the two orbital-month numbers.

2. Take the shortest distance around the twelve-position circle.

3. The result is a gap from 0 to 6.

4. Multiply mentally by about 30° if an angular picture is useful.

5. Combine that broad geometry with knowledge of the two worlds' orbital scale.

6. Treat gap 0 as broadly towards the favourable end of the pair's range cycle and gap 6 as broadly towards the unfavourable end.

7. Use an ephemeris whenever the actual number matters.

That is all the heuristic is. And that is probably why it may be useful.

It compresses a moving Solar System into something a human being can carry around without doing astronomy in their head.

Destination gives the scale.

Month gap gives the broad geometry.

Software gives the precision.