Tidally Locked Habitable Ring Calculator

What this calculator estimates

Tidally locked planets keep one hemisphere facing their star (permanent day) while the opposite hemisphere remains in perpetual night. The strongest heating occurs at the substellar point (the point directly under the star). Moving away from that point toward the terminator (the day–night boundary), incoming starlight arrives at a lower angle and the absorbed energy drops. The result is often imagined as a circumplanetary “twilight” band where temperatures may fall in a comfortable range for liquid water, agriculture, or settlement.

This calculator estimates the angular width, surface distance, and surface area of the region on the dayside where the surface temperature lies between your chosen minimum and maximum comfortable temperatures. It uses a deliberately simplified energy-balance relation intended for quick exploration and worldbuilding rather than climate prediction.

Model and variable definitions

Inputs:

• Planet radius R (km): the planetary radius used to convert angles into kilometers and to compute surface area.
• Substellar temperature T_s (K): the surface temperature at the substellar point (maximum in this simple model).
• Minimum comfortable temperature T_min (K): the cold edge of the desired habitable/usable range.
• Maximum comfortable temperature T_max (K): the hot edge of the desired habitable/usable range.

Geometry:

• θ is the zenith angle from the substellar point along the surface (0° at the substellar point; 90° at the terminator). Over the dayside, cos(θ) ranges from 1 down to 0.

Temperature–angle relation (cosine-to-the-quarter law)

A common first-order approximation for radiative equilibrium on the illuminated hemisphere is that absorbed flux scales with cos(θ), while equilibrium temperature scales with the fourth root of flux. This yields:

Solving for angle at a chosen temperature T:

cos(θ) = (T / T_s)^4

Then:

• Hot-edge angle (where the surface cools down to T_max): θ_hot = arccos((Tmax/Ts)^4)
• Cold-edge angle (where it cools down to T_min): θcold = arccos((Tmin/Ts)^4)

The ring’s angular thickness on the dayside is Δθ = θcold - θhot (in radians or degrees, depending on how you report it). The approximate surface width along the ground is:

width_km = R × Δθ (radians)

Surface area of the habitable band

On a sphere, the area between two zenith angles measured from the substellar point is:

A = 2π R^2 (cos(θ_hot) - cos(θcold))

This is the area of the dayside band whose temperatures fall between T_max and T_min under the model assumptions.

How to interpret the results

• If Tmax > Ts, the model cannot produce temperatures hotter than T_s anywhere; effectively the “too hot” boundary collapses to the substellar point (angle 0°). In practice, you should set T_max ≤ T_s if you want a meaningful hot-edge boundary.
• If Tmin is very low, the cold-edge angle can approach the terminator (near 90°). When T_min is at or below the model’s terminator temperature (which tends toward 0 K in this idealized form without heat transport), the band can extend all the way to the terminator.
• A wider ring means more surface real estate at your target temperatures. A larger planet increases both width (km) and area roughly with R and R², respectively, even if the angular band is unchanged.

Worked example

Suppose a tidally locked rocky planet has:

• R = 6371 km
• T_s = 400 K
• T_max = 310 K
• T_min = 270 K

Compute cosine values:

• cos(θ_hot) = (310/400)^4 ≈ 0.361
• cos(θcold) = (270/400)^4 ≈ 0.208

Angles (degrees):

• θ_hot ≈ arccos(0.361) ≈ 68.8°
• θcold ≈ arccos(0.208) ≈ 78.0°

Angular thickness: Δθ ≈ 9.2° ≈ 0.161 rad. Surface width: width ≈ 6371 × 0.161 ≈ 1030 km.

Area:

A = 2π R^2 (0.361 - 0.208) ≈ 2π (6371^2) (0.153) ≈ 39 million km^2 (order-of-magnitude).

Interpretation: under this simplified model, a sizable belt near the terminator stays within 270–310 K. In a worldbuilding context, that could support a broad “ring civilization” with large agricultural area—if the atmospheric and circulation assumptions below are reasonably satisfied.

Comparison table: what each output tells you

Model assumptions and limitations (important)

• Radiative equilibrium, simple insolation geometry: temperature scales as the fourth root of cos(θ). Real planets include greenhouse effects and wavelength-dependent absorption.
• No clouds, no albedo variation: reflectivity is treated as implicitly constant. Clouds can strongly cool the substellar region and warm the terminator depending on circulation.
• No explicit atmospheric/ocean heat transport: the formula is often used as a baseline, but real tidally locked climates can advect heat to the nightside and flatten temperature contrasts, widening or shifting the “comfortable” zone.
• Dayside-only interpretation: this band is computed on the illuminated hemisphere (0–90° from substellar). Nightside habitability is not modeled.
• Boundary conditions: inputs that imply (T/Ts)^4 > 1 or < 0 are physically invalid for this model and should be treated as “no solution” or clamped boundaries.
• Topography and seasons ignored: elevation, oceans/continents, obliquity, eccentricity, and time variability can dominate local habitability.

Reference (conceptual)

The cosine-based insolation scaling and T ∝ F^1/4 radiative equilibrium relationship are standard tools in introductory planetary energy balance modeling and are commonly used as first approximations in exoplanet climate discussions.