Abstract
A spacecraft is reported to have “left the solar system”. This paper asks which edge it left——the answer is that there are five edges, and it was the second from the inside. No new mathematical theorem and no new law is claimed.
First, the five definitions run from $50$ AU to $\mathbf{255302}$ AU, a factor of $\mathbf{5106.0345}$ (Section 2).
Second, this is the core of the paper. The five are not a ranking; each names a different thing that ends (Section 3).
Third, the outermost, the Hill radius, is computed here from the rotation of the Galaxy. It comes to $\mathbf{1.237738}$ pc, inside the usual figure of about $1$ pc (Section 2).
Fourth, the Sun’s gravitational domain reaches $\mathbf{95.07}$ per cent of the way to Proxima (Section 4).
Fifth, light takes $16.8553$ hours to the heliopause and $4.0370$ years to the Hill radius (Section 4).
Sixth, the separator is what it is that ends (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the Hill radius, the heliopause, the Oort cloud, the Kuiper belt, and the galactic rotation curve are all standard. We do not build celestial mechanics——the three-body problem and the derivation of the Hill radius are not entered. The formula is used as given. We do not treat solar-wind physics——the termination shock, the structure of the heliosheath, and field winding are not entered. They are mentioned and no more. We do not adjudicate whether the Oort cloud exists——that it has never been observed directly is stated plainly. The figures are the accepted estimates. We do not treat interstellar space——the local interstellar cloud and the interstellar magnetic field are not entered. We do not treat mission operations——which craft measured what and when is not entered in detail. Relation to earlier papers: Paper 306 showed how many “base” units there are to be a promise——the five here are promises too, but none of them is arbitrary; each names where a different physical quantity ends. That there are several promises and that one of them is right are different matters. Paper 297 showed cosmic distance to be a ladder of seven rungs——there one quantity, distance, was measured seven ways; here one place, the edge, is set by five criteria. Many methods and many definitions are not the same thing. Paper 300 showed whether two things share a root is decidable——by that test the five here have distinct roots from one another. Paper 328 showed that the same ellipse being three months out was evidence of a different cause——there and here alike, two or more separate quantities ride on one astronomical object. What is added is putting the five in one table in one unit, computing the Hill radius from the galactic rotation curve and testing the tool on the Earth’s Hill radius, giving the jumps between neighbours to show the spacing is uneven, and putting the separator on what it is that ends.
1. Notation and the Question
Setting: The phrase “the edge of the solar system” is used in ordinary speech and in the press. But what counts as the edge has never been settled to one thing.
Observation 1.1 Voyager 1 was reported to have left the solar system in $2012$. By another criterion the same craft has not left——not a contradiction; a different criterion.
Observation 1.2 The way of measuring is fixed first. All distances in astronomical units, with $1$ AU $=1.495978707\times10^{11}$ m——ratios are invisible when the units differ.
Observation 1.3 Four of the five are accepted figures; one is computed here. The Hill radius alone is derived from galactic rotation——what is quoted and what is computed are written apart.
Question: How many edges are there, and how far apart do they lie?
2. The Five Edges in One Unit
Calculation 2.1: Compute the Hill radius from the galactic rotation curve (computed in this paper). Taking the Sun on a circular orbit at $R_0=8$ kpc with $v_c=220$ km/s, the enclosed mass is $M=v_c^2R_0/G$ and the Hill radius is $R_H=R_0(M_\odot/3M)^{1/3}$.
| Quantity | Value |
|---|---|
| Enclosed mass $M(<8 \text{kpc})$ | $\mathbf{9.0004\times10^{10}} M_\odot$ |
| Hill radius | $\mathbf{1.237738}$ pc |
| the same | $\mathbf{255301.7}$ AU |
| the same | $\mathbf{4.0370}$ light years |
Observation 2.2 Two checks. (a) The enclosed mass $9.0004\times10^{10} M_\odot$ falls in the usual decade of $10^{11} M_\odot$. (b) The same formula applied to the Earth gives $0.010003$ AU $=1496433$ km, agreeing with the usual figure of about $1.5$ million km——the tool is tried on another body whose answer is known.
Calculation 2.3: The five in one unit (all but the Hill radius are accepted figures).
| Definition | Distance [AU] | What ends there |
|---|---|---|
| Outer rim of the Kuiper belt | $50$ | the distribution of icy bodies |
| Heliopause | $121.6$ | the solar wind loses to the interstellar medium |
| Oort cloud, inner edge | $2000$ | the reservoir of long-period comets begins |
| Oort cloud, outer edge | $100000$ | the same, outward |
| Hill radius (computed here) | $\mathbf{255301.7}$ | solar gravity loses to the Galaxy |
Observation 2.4 Smallest to largest is a factor of $\mathbf{5106.0345}$. One phrase, “the edge of the solar system”——covering three and a half decades.
Calculation 2.5: The jumps between neighbours (computed in this paper).
| Interval | Ratio |
|---|---|
| Kuiper rim $\to$ heliopause | $2.4320$ |
| Heliopause $\to$ Oort inner edge | $16.4474$ |
| Oort inner $\to$ outer edge | $\mathbf{50.0000}$ |
| Oort outer edge $\to$ Hill radius | $2.5530$ |
Observation 2.6 The jumps are uneven. The largest, $50.0000$, and the smallest, $2.4320$, differ by $20.5592$——the five are not evenly spaced. The Oort cloud alone spans nearly two decades.
3. The Core: They Are Not a Ranking but Places Where Different Quantities End
Setting: Sort the five by what it is that ends.
| Definition | What ends | Kind |
|---|---|---|
| Kuiper rim | the distribution of bodies | matter |
| Heliopause | the pressure of the solar wind | fluid |
| Oort cloud | the reservoir of comets | matter |
| Hill radius | the dominance of solar gravity | force |
Observation 3.1 This is the core of the paper. The five do not differ in “how far one counts”——the physical quantity that ends is a different quantity in each case. A distribution of matter, a fluid pressure and a dominance of force cannot be converted into one another.
Observation 3.2 Apply the criterion of Paper 300.
| Criterion | Result |
|---|---|
| Is there a correspondence preserving an identity | No. Pressure balance and gravitational balance are different equations |
| Does the same quantity appear | No. $\rho v^2$ against $GM/r^2$ |
| Do they keep agreeing as variables move | No. A stronger wind moves the heliopause alone |
Observation 3.3 All three rows point to distinct roots. The heliopause moves year by year with the solar cycle; the Hill radius is set by the mass distribution of the Galaxy——move one and the other stays.
Observation 3.4 So “which is the real edge” is not a well-posed question. The question to ask is the edge of what——the position Paper 306 took on the count of base units, except that here there are five promises and not one of them is arbitrary.
Observation 3.5 The difference from Paper 297, stated. Paper 297 wrote that the distance ladder has seven rungs——there one and the same quantity, distance, was measured seven ways. Here the quantity being measured differs in all five. Many methods and many definitions are not the same thing.
Observation 3.6 To be stated honestly——the Oort cloud has never been observed directly. It is a structure inferred from the statistics of long-period comet orbits——two of the five (its edges) rest on weaker evidence than the other three. They are in the same table and are not equally certain.
4. How Long Light Takes
Calculation 4.1: Light travel time to each of the five (computed in this paper).
| Definition | Distance [AU] | Light travel time |
|---|---|---|
| Kuiper rim | $50$ | $6.9306$ hours |
| Heliopause | $121.6$ | $\mathbf{16.8553}$ hours |
| Oort inner edge | $2000$ | $277.2249$ hours ($11.6$ days) |
| Oort outer edge | $100000$ | $1.5813$ years |
| Hill radius | $255301.7$ | $\mathbf{4.0370}$ years |
Observation 4.2 Light reaches the heliopause in under a day. $16.8553$ hours——a distance Voyager took $35$ years to cover, light crosses while one sleeps.
Observation 4.3 To the Hill radius it takes $4.0370$ years. Still inside the same “solar system”——in time, the factor of $5106$ becomes something one can feel.
Calculation 4.4: Compare with the nearest star (computed in this paper).
| Quantity | Value |
|---|---|
| Hill radius of the Sun | $\mathbf{4.0370}$ light years |
| Distance to Proxima Centauri | $4.2465$ light years |
| Ratio | $\mathbf{0.9507}$ |
Observation 4.5 The Sun’s gravitational domain reaches $95.07$ per cent of the way to the nearest star. Which is to say there is almost no gap in interstellar space——the outside of this system and the outside of the next nearly touch.
Observation 4.6 To be stated honestly——the Hill radius is a domain against the galactic tide, not against the neighbouring star. The figure $0.9507$ places two values from different criteria side by side——the gravitational boundary against a neighbouring star, an equipotential surface, is another calculation. It is not entered here.
5. The Separator: What Is It That Ends?
Observation 5.1 The separator——whether what ends at that boundary is a distribution of matter, a fluid pressure, or a dominance of force.
| What ends | Set by | Moves in time | |
|---|---|---|---|
| Distribution | number density of bodies | formation history | no |
| Pressure | the solar wind | solar activity | yes |
| Force | gravitational dominance | the mass of the Galaxy | no |
Observation 5.2 The last column splits the five in two. Only the heliopause moves year by year with the solar cycle——“the day it left the solar system” means something only at this boundary. The other four have no such day.
Observation 5.3 The separator can be moved. Stop the solar wind and the heliopause vanishes; change the mass of the Galaxy and only the Hill radius moves——the five can be moved independently, and so they are five independent things.
Observation 5.4 The press phrasing never says which edge. Saying “it left the solar system” leaves it unclear whether $121.6$ AU or $255302$ AU is meant——two figures a factor of $2099.5206$ apart go under one phrase.
Observation 5.5 To be stated honestly——there is no need to settle on one “real” edge. What is needed is to say each time which edge is meant——as with the count of units in Paper 306, the promise is not the trouble; leaving the promise unstated is.
Observation 5.6 One last thing——only the outermost edge is not fixed by any property of the Sun. What sets the Hill radius is how hard the Galaxy pulls on the Sun——the outermost outline of the solar system was not drawn by the Sun; it was handed in from outside.
6. The Census and the Close
- Five edges from $50$ to $255301.7$ AU (Calculation 2.3): a factor of $5106.0345$.
- Hill radius $1.237738$ pc (Calculation 2.1): from galactic rotation.
- Earth’s Hill radius $1496433$ km (Observation 2.2): same formula, agrees.
- Jumps from $2.4320$ to $50.0000$ (Calculation 2.5): not evenly spaced.
- A different quantity ends in each (Observation 3.1): distribution, pressure, force.
- Distinct roots on all three criteria (Observation 3.2): not interconvertible.
- $16.8553$ hours to the heliopause (Calculation 4.1): under a day.
- $4.0370$ years to the Hill radius (Calculation 4.1): inside the same system.
- $95.07$ per cent of the way to Proxima (Calculation 4.4): almost no gap.
- Only the heliopause moves (Observation 5.2): only it has a date.
Closing: One phrase, “the edge of the solar system”, names five different places——the outer rim of the Kuiper belt at $50$ AU, the heliopause at $121.6$ AU, the inner and outer edges of the Oort cloud at $2000$ and $100000$ AU, and the Hill radius computed here from galactic rotation at $\mathbf{255301.7}$ AU ($\mathbf{1.237738}$ pc). Smallest to largest is a factor of $\mathbf{5106.0345}$, one phrase covering three and a half decades. And the five are not a ranking——the physical quantity that ends is different in each. The Kuiper rim ends a distribution of bodies, the heliopause a pressure, the Hill radius a dominance of gravity, and these cannot be converted into one another. All three criteria of Paper 300 point to distinct roots. So “which is the real edge” is not a well-posed question——the question is the edge of what. It is the position Paper 306 took on the count of base units, except that here there are five promises and not one of them is arbitrary. In light travel time the difference becomes something one can feel——the heliopause is $\mathbf{16.8553}$ hours away, so a distance Voyager took $35$ years to cover, light crosses while one sleeps. The Hill radius is $\mathbf{4.0370}$ years away——still inside the same system. And that Hill radius reaches $\mathbf{95.07}$ per cent of the way to the nearest star——Proxima Centauri lies at $4.2465$ light years. There is almost no gap in interstellar space. Though the Hill radius is a domain against the galactic tide and not against the neighbouring star, so this ratio places two different criteria side by side. One thing separates them——whether what ends is a distribution of matter, a fluid pressure, or a dominance of force. Only the heliopause moves year by year with the solar cycle——“the day it left the solar system” means something at that boundary alone; the other four have no such day. When the press says “it left the solar system”, $121.6$ AU and $255302$ AU, a factor of $2099.5206$ apart, go under one phrase. To be said honestly——the Oort cloud has never been observed directly. It is inferred from the statistics of long-period comet orbits, and two of the five rest on weaker evidence than the other three. Nor is there any need to settle on one “real” edge. What is needed is to say each time which edge is meant. One last thing——only the outermost edge is not fixed by any property of the Sun. What sets the Hill radius is how hard the Galaxy pulls on the Sun. The outermost outline of the solar system was not drawn by the Sun; it was handed in from outside.
References
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- Yamagishi, Y. (2026). Cosmic Distance Is a Ladder of Seven Rungs, and Only the First Is Measured Directly [Paper 297]. Zenodo. 10.5281/zenodo.22171788
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). How Many “Base” Units There Are Is a Promise [Paper 306]. Zenodo. 10.5281/zenodo.22172149
- Yamagishi, Y. (2026). That the Same Ellipse Was Three Months Out Was the Evidence of a Different Cause [Paper 328]. Zenodo. 10.5281/zenodo.22210340
Reference
Yamagishi, Y. (2026). The Edge of the Solar System Is Five Edges, and the Innermost and Outermost Differ by a Factor of $5106$ [Paper 337]. YOI: https://www.yamagishiyuuki.com/yoi/337
DOI: 10.5281/zenodo.22681940
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 337. This paper is a preprint and has not been peer reviewed.