Formula used
Use when luminosity and radius are known or estimated.
Useful for radiative surface energy budgets.
D controls heat redistribution: 4 full, 2 dayside, 1 substellar.
How to use this calculator
- Select a model matching your available measurements.
- Pick the variable you want to solve for.
- Enter inputs using consistent units and realistic values.
- Adjust emissivity and constants for specialized scenarios.
- Press Calculate to see results above the form.
- Use CSV or PDF buttons to export your latest run.
Example data table
These examples show typical inputs and approximate temperatures.
| Scenario | Inputs | Output |
|---|---|---|
| Sun (star) | L = 3.828×10²⁶ W, R = 6.9634×10⁸ m, ε = 1 | T ≈ 5772 K |
| Earth-like (planet) | S = 1361 W/m², A = 0.30, D = 4, ε = 1, ΔT = 33 K | T ≈ 288 K |
| Hot exoplanet dayside | S = 8000 W/m², A = 0.10, D = 2, ε = 0.9, ΔT = 0 K | T ≈ 1800 K |
Article
1) Why “effective temperature” matters
Effective temperature is the blackbody-equivalent temperature that reproduces an object’s total radiated power. It compresses complex spectra and surface physics into one comparable number. In stellar work it links luminosity, radius, and emission efficiency; in planetary work it links absorbed sunlight, reflection, and heat transport.
2) Stefan–Boltzmann scaling and units
Thermal emission rises steeply with temperature through the Stefan–Boltzmann law, where radiative flux scales as T4. The calculator uses σ = 5.670374419×10−8 W·m−2·K−4. This fourth‑power dependence means small temperature errors can imply large power differences.
3) Emissivity as a real‑surface correction
Many surfaces are “grey,” emitting less than an ideal blackbody. Emissivity ε (0–1) scales radiated flux and shifts inferred temperatures. For the same flux, lowering ε increases the temperature estimate by a factor of ε−1/4. That mild exponent is helpful, but ε still matters for metals, dusty regions, and atmospheres.
4) Star model from luminosity and radius
When luminosity L and radius R are known, the model uses T = (L / (4πR²εσ))1/4. With solar values (L≈3.828×1026 W and R≈6.9634×108 m), the computed effective temperature is near 5770 K. This is a global radiative value, not a photospheric line‑formation temperature.
5) Star model from surface flux
If you have measured or simulated surface flux F, the flux model uses T = (F /(εσ))1/4. This approach is common in thermal engineering and climate‑style energy budgets. It avoids geometric factors and focuses on local emission, making it suitable for mapped surfaces or boundary‑condition checks.
6) Planet equilibrium temperature with albedo
For planets, absorbed shortwave power depends on Bond albedo A, which is the fraction of incident energy reflected over all angles and wavelengths. The calculator uses Teq = (((1−A)S)/(Dεσ))1/4, then optionally adds a greenhouse offset ΔT to represent atmospheric warming.
7) Redistribution divisor and day–night physics
The divisor D captures how absorbed energy is spread before re‑emission. D≈4 approximates efficient global redistribution, D≈2 represents a dayside mean, and D≈1 approximates a hot substellar point. For close‑in exoplanets, changing D can shift temperatures by hundreds of kelvin at high S.
8) Interpreting outputs and uncertainty
Use Kelvin for physics consistency, then convert to °C/°F for communication. Compare scenarios by keeping ε, A, and D explicit so assumptions stay visible in exports. Typical uncertainty drivers are albedo (often ±0.05–0.2), emissivity, and redistribution; propagate them by re‑running the calculator with plausible bounds.
FAQs
1) Is effective temperature the same as surface temperature?
No. It is a radiative equivalent that matches total emitted power. Real surfaces have temperature variations, spectral features, and atmospheric effects, so local or measured surface temperatures can differ from the effective value.
2) What emissivity should I use if I do not know it?
For first estimates, ε = 1 is common. Polished metals can be 0.05–0.3, many rocks and paints 0.85–0.98, and dusty or rough surfaces often approach 0.9–1. Adjust ε to match your material or model assumptions.
3) Why does the planet model use D = 4 for “full redistribution”?
Absorption is over a cross‑section (πR²) while emission is over the full sphere (4πR²). That geometry produces a factor near 4 when heat is efficiently spread before being radiated away.
4) How does albedo change temperature?
Temperature scales with (1−A)1/4. Increasing albedo reduces absorbed energy and lowers temperature. For example, changing A from 0.3 to 0.4 decreases (1−A) from 0.7 to 0.6, lowering T by a few percent.
5) What does the greenhouse offset ΔT represent?
It is a simple additive correction for atmospheric warming beyond the pure radiative equilibrium estimate. It is not a full radiative‑transfer solution, but it helps match observed mean temperatures when greenhouse trapping is significant.
6) Can I solve for luminosity or radius using this tool?
Yes. In the star luminosity–radius model, choose “Solve for” L or R and provide the target temperature plus the remaining variable. The tool then inverts the Stefan–Boltzmann relationship consistently.
7) Why are results shown in scientific notation?
Astrophysical and radiative values can span many orders of magnitude. Scientific notation reduces rounding ambiguity and keeps exports stable. You can copy the outputs directly into reports, spreadsheets, or simulation inputs.