Abstract
The habitable zone is spoken of as a property of a star. This paper asks where its boundary comes from——the answer is not from the star but from assumptions about the planet’s albedo and atmosphere. No new mathematical theorem and no new law is claimed.
First, at albedo $0.30$ Earth’s equilibrium temperature is $\mathbf{254.5781}$ K, $18.57$ degrees below freezing (Section 2).
Second, this is the core of the paper. On that assumption the “freezing-point distance” is $\mathbf{0.8686}$ AU, and Earth falls outside the boundary (Section 3).
Third, set the albedo to $0$ and the boundary becomes $1.0382$ AU, putting Earth inside. One assumption reverses the conclusion (Section 3).
Fourth, from $0$ to $0.75$ the boundary moves $0.5191$ AU, a ratio of exactly $\mathbf{2}$ (Section 3).
Fifth, Earth is actually at $288$ K because of $\mathbf{33.4219}$ K of greenhouse effect (Section 4).
Sixth, the separator is whether the atmosphere is in the formula, not the star and not the distance (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——equilibrium temperature, albedo, the greenhouse effect, and the habitable zone are all standard. We do not build climatology——radiative-convective equilibrium, the runaway greenhouse, and regulation by the carbon cycle are not entered. They are named and no more. We evaluate no planets——whether any particular exoplanet is habitable is not judged. We do not define life——conditions other than liquid water are not the subject. What is treated here is a temperature formula. Stellar evolution is not treated——main-sequence luminosity change and the continuously habitable zone are named and no more. Tides are not treated——synchronous rotation and tidal heating are not the subject. Relation to earlier papers: Paper 358 showed the “depth” of depth of field to be a convention rather than a measurement——there the convention was the circle of confusion; here it is the albedo. Paper 352 showed a transition temperature moving with the speed of measurement——there too a quantity wearing the face of a constant was conditional. Paper 300 showed whether two things share a root is decidable——by that test, equilibrium temperature and surface temperature have distinct roots. Paper 353 showed a claim turning false when its condition is dropped——here dropping it classifies Earth as uninhabitable. What is added is giving equilibrium temperature and boundary distance at five albedos, showing that the standard assumption excludes Earth itself, showing the boundary’s ratio to be exactly $2$, checking the $33.4219$ K greenhouse contribution against the accepted figure, and putting the separator on whether the atmosphere is in the formula.
1. Notation and the Question
Setting: The equilibrium temperature is the temperature at which a sphere of albedo $A$ balances an irradiance $S$.
$$T_{\text{eq}}=\left[\frac{S(1-A)}{4\sigma}\right]^{1/4}, \qquad S=\frac{S_0}{d^2},\qquad S_0=1361\ \mathrm{W/m^2}$$Here $d$ is distance in astronomical units and $\sigma$ the Stefan--Boltzmann constant.
Observation 1.1 There is no atmosphere in this formula. Only the radiation received and the fraction reflected——no layer that absorbs and re-emits is counted.
Observation 1.2 Albedo is a property of the planet. Not of the star——so a “zone” cannot be drawn from the star alone.
Observation 1.3 And albedo is itself an outcome of that planet’s climate. Ice raises it, fewer clouds lower it——what should be an input depends on an output.
Question: What fixes the boundary of a habitable zone?
2. Earth’s Equilibrium Temperature Is Below Freezing
Calculation 2.1: Equilibrium temperature at $1$ AU against albedo (computed in this paper).
| Albedo $A$ | $T_{\text{eq}}$ at $1$ AU [K] | Celsius |
|---|---|---|
| $0.00$ | $\mathbf{278.3214}$ | $+5.17$ |
| $0.10$ | $271.0861$ | $-2.06$ |
| $\mathbf{0.30}$ | $\mathbf{254.5781}$ | $\mathbf{-18.57}$ |
| $0.50$ | $234.0395$ | $-39.11$ |
| $0.75$ | $196.8029$ | $-76.35$ |
Observation 2.2 Earth’s actual albedo is about $0.30$. The equilibrium temperature there is $254.5781$ K——$-18.57$ degrees Celsius, well below freezing.
Observation 2.3 This matches the accepted value. Earth’s equilibrium temperature is usually given as about $255$ K——the $254.5781$ K here differs by only $0.42$ K. Formula and constants are right.
Observation 2.4 Albedo acts through a fourth root. Taking $A$ from $0$ to $0.75$ quarters $(1-A)$, but the temperature falls only by $\sqrt{2}$——from $278.3214$ to $196.8029$, a ratio of $1.4142$.
Observation 2.5 And still it moves $81.5$ K. Blunted by the fourth root and this wide even so——the albedo assumption governs the temperature conclusion.
3. The Boundary Moves with the Assumption
Calculation 3.1: Solving backwards for the distance at which $T_{\text{eq}}=273.15$ K (computed in this paper).
| Albedo $A$ | Boundary distance [AU] | Earth ($1$ AU) is |
|---|---|---|
| $0.00$ | $\mathbf{1.0382}$ | inside (habitable) |
| $0.10$ | $0.9849$ | outside |
| $\mathbf{0.30}$ | $\mathbf{0.8686}$ | outside (uninhabitable) |
| $0.50$ | $0.7341$ | outside |
| $0.75$ | $0.5191$ | outside |
Observation 3.2 This is the core of the paper. With the standard $A=0.30$ the boundary is $0.8686$ AU and Earth falls outside it——the one known instance of life is classified as uninhabitable.
Observation 3.3 Setting $A=0$ reverses it. The boundary becomes $1.0382$ AU and Earth is inside——one changed assumption about the planet flips the conclusion.
Observation 3.4 The reversal happens below $A=0.10$. At $A=0.10$ the boundary is $0.9849$ AU, still inside Earth’s orbit——so Earth qualifies only for $A$ under about $0.1$, far from the actual $0.30$.
Observation 3.5 The boundary’s motion has a closed form. $d\propto\sqrt{1-A}$, so the ratio between $A=0$ and $A=0.75$ is $\sqrt{1/0.25}=2$——measured, exactly $2.0000$, opening a span of $0.5191$ AU.
Observation 3.6 The inversion closes. Putting the resulting $0.8686$ AU back into the formula returns $273.15000000$ K——this is not an arithmetic error.
4. The Missing $33.4219$ K
Observation 4.1 Earth’s actual mean surface temperature is $288$ K. The gap from the equilibrium value $254.5781$ K is $\mathbf{33.4219}$ K——that is the greenhouse contribution.
Observation 4.2 It falls in the accepted $30$ to $35$ K. $33.4219$ K——agreement with an independently known value, and a check on the computation here.
Observation 4.3 So what the formula omits is the atmosphere. The equilibrium formula has no absorbing layer——$33.4219$ K is the size of what was left out.
Observation 4.4 Venus is the extreme case. Its albedo is high and its equilibrium temperature lower than Earth’s, yet its surface exceeds $700$ K——the ordering by equilibrium temperature does not match the ordering by surface temperature. This paper has not computed those numbers and states the fact only.
Observation 4.5 Distinct roots by the criterion of Paper 300. Equilibrium temperature is fixed by a radiation balance; surface temperature by that balance and the atmosphere’s optical depth——both are “temperature”, and different things fix them.
5. The Separator: Is the Atmosphere in the Formula?
Observation 5.1 The separator——whether the atmosphere is in the formula; not the star and not the distance.
| What enters | Answer for Earth | Classifies Earth | |
|---|---|---|---|
| Equilibrium temperature | $S$ and $A$ only | $\mathbf{254.5781}$ K | wrongly |
| Surface temperature | plus optical depth | $\mathbf{288}$ K | correctly |
Observation 5.2 The rightmost column is the whole verdict. Tried on the one known instance, the top row fails——the same discipline as the corpus’s scanning rule: test on a case that ought to hit.
Observation 5.3 The separator can be moved. Add a fixed greenhouse increment and the boundary moves outward——but how much to add differs by planet, and that is itself a new assumption.
Observation 5.4 The assumption does not vanish; it changes place. Assume an albedo, or assume a greenhouse effect——the same shape as the circle-of-confusion convention merely changing place in Paper 358.
Observation 5.5 To be stated honestly——actual habitable-zone research does not draw boundaries from equilibrium temperature alone. Models including radiative-convective equilibrium and the runaway greenhouse are used——what this paper shows is that the most naive way of drawing it fails on the one known instance.
Observation 5.6 One last thing——a classifier should be tried on a case whose answer is known. Tried on Earth, classification by equilibrium temperature fails——apply it to exoplanets without trying it, and the failure goes unnoticed.
6. The Census and the Close
- $T_{\text{eq}}=254.5781$ K at $A=0.30$ (Calculation 2.1): $-18.57$ degrees Celsius.
- Only $0.42$ K from the accepted $255$ K (Observation 2.3): the formula is right.
- $278.3214$ K at $A=0$ (Calculation 2.1): a span of $81.5$ K.
- Boundary $0.8686$ AU at $A=0.30$ (Calculation 3.1): Earth outside.
- Boundary $1.0382$ AU at $A=0$ (Calculation 3.1): Earth inside; the conclusion reverses.
- Ratio exactly $2.0000$ (Observation 3.5): $d\propto\sqrt{1-A}$.
- Back-substitution gives $273.15000000$ K (Observation 3.6): the formula closes.
- Greenhouse contribution $33.4219$ K (Observation 4.1): inside the accepted $30$--$35$ K.
- Albedo is itself a climate outcome (Observation 1.3): an input depending on an output.
- Test on a known case (Observation 5.2): tried on Earth, it fails.
Closing: The habitable zone is spoken of as a property of a star——but its boundary cannot be drawn from the star alone. The equilibrium-temperature formula has no atmosphere in it——only the radiation received and the fraction reflected, and albedo is a property of the planet — indeed an outcome of that planet’s climate. What should be an input depends on an output. And the standard assumption puts Earth itself outside——at $A=0.30$ Earth’s equilibrium temperature is $\mathbf{254.5781}$ K, $-18.57$ degrees Celsius (only $0.42$ K from the accepted $255$ K). On that assumption the “freezing-point distance” is $\mathbf{0.8686}$ AU and Earth falls outside the boundary. The one known instance of life is classified as uninhabitable. One changed assumption reverses the conclusion——set $A=0$ and the boundary becomes $1.0382$ AU, putting Earth inside. At $A=0.10$ it is still $0.9849$ AU, so Earth qualifies only for $A$ under about $0.1$ — far from the actual $0.30$. Since $d\propto\sqrt{1-A}$, the ratio between $A=0$ and $A=0.75$ is exactly $\mathbf{2.0000}$, a span of $0.5191$ AU. What is missing is $\mathbf{33.4219}$ K——the gap between Earth’s actual $288$ K and its equilibrium value, inside the accepted $30$ to $35$ K. It is the greenhouse contribution, and equally the size of what the formula left out. By the criterion of Paper 300, equilibrium temperature and surface temperature have distinct roots: both are “temperature” and different things fix them. One thing separates them——whether the atmosphere is in the formula; not the star and not the distance. And the assumption does not vanish, it changes place——assume an albedo, or assume a greenhouse effect: the same shape as the circle-of-confusion convention merely changing place in Paper 358. To be said honestly——actual habitable-zone research does not draw boundaries from equilibrium temperature alone; models including radiative-convective equilibrium and the runaway greenhouse are used. What this paper shows is that the most naive way of drawing it fails on the one known instance. One last thing——a classifier should be tried on a case whose answer is known. Tried on Earth, classification by equilibrium temperature fails. Apply it to exoplanets without trying it, and the failure goes unnoticed.
References
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- Kopparapu, R. K., et al. (2013). Habitable Zones around Main-sequence Stars: New Estimates. The Astrophysical Journal.
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- Pierrehumbert, R. T. (2010). Principles of Planetary Climate. Cambridge: Cambridge University Press.
- Kopp, G., Lean, J. L. (2011). A new, lower value of total solar irradiance. Geophysical Research Letters.
- 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). The Glass Transition Temperature Moves with the Speed of Measurement [Paper 352]. Zenodo. 10.5281/zenodo.22695723
- Yamagishi, Y. (2026). Adding Noise Can Improve Detection — But “More Is Better” Is Not the Rule [Paper 353]. Zenodo. 10.5281/zenodo.22696835
- Yamagishi, Y. (2026). The “Depth” of Depth of Field Is a Convention, Not a Measurement [Paper 358]. Zenodo. 10.5281/zenodo.22697216
Reference
Yamagishi, Y. (2026). A Habitable Zone Drawn from Equilibrium Temperature Puts Earth Outside It [Paper 361]. YOI: https://www.yamagishiyuuki.com/yoi/361
DOI: 10.5281/zenodo.22697436
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 361. This paper is a preprint and has not been peer reviewed.