ThermoCalc

Stefan-Boltzmann Calculator

Every object radiates heat — the Stefan-Boltzmann law says how much, and it scales with T⁴.

Temperature (K)

Surface area (m²)

Emissivity (0–1)

Total power (W)

63,200,699.735

P = εσAT⁴

Radiant flux (W/m²): 63,200,699.735

What you entered

Temperature (K): 5778Surface area (m²): 1Emissivity (0–1): 1

1
Radiant flux: j = εσT⁴
1 × 5.6704e-8 × 5778⁴= 6.3201e+7 W/m²
2
Total power: P = j × A
6.3201e+7 × 1= 6.3201e+7 W

Result

Total power (W): 63,200,699.735

A body at 5778 K (ε = 1) with 1 m² surface area radiates 6.3201e+7 W.

Thermal radiation from a hot surface

Radiated power is emissivity times the Stefan-Boltzmann constant times area times temperature to the fourth power.

Temperature (K)	Area (m2)	Emissivity	Power (W)	Flux (W/m2)
273	1	1	3.1497e+2	3.1497e+2
300	1	1	4.5930e+2	4.5930e+2
310	1.8	0.98	9.2376e+2	5.1320e+2
1000	0.5	0.9	2.5517e+4	5.1033e+4
5778	1	1	6.3201e+7	6.3201e+7

The fourth-power dependence is dramatic: the 273 K and 1000 K rows differ by less than a factor of four in temperature but by more than 160 times in flux. The third row is roughly a human body at skin temperature radiating about 920 W, though the net loss is far smaller because the surroundings radiate back. The last row is the Sun's surface at 5,778 K. Emissivity ranges from near 0 for polished metal to near 1 for matt black, which is why radiators are painted dark and spacecraft are wrapped in reflective foil.

The dramatic T⁴ dependence

Radiated power scales with the fourth power of absolute temperature — double the temperature and radiation increases 16-fold. This is why a glowing-hot iron bar at 1000 K radiates roughly 16 times more than the same bar at 500 K, and why the Sun (5778 K) emits an enormous amount of energy.

Emissivity: real surfaces vs. ideal blackbodies

A perfect blackbody has emissivity ε = 1 and radiates the theoretical maximum. Real surfaces have ε < 1: polished aluminum is about 0.05 (poor radiator), while matte black paint is about 0.95 (nearly ideal). This is why spacecraft use different surface coatings to manage thermal radiation.

Frequently asked questions

The Sun's surface temperature is 5,778 K and its radius is 6.96×10⁸ m. What's its total radiated power?

P = εσAT⁴ = 1 × 5.67×10⁻⁸ × 4π(6.96×10⁸)² × 5778⁴ ≈ 3.85×10²⁶ W. That's 385 yottawatts — Earth intercepts only about 1.74×10¹⁷ W (0.000000045%) of this, yet it powers all life and weather on our planet.

A glowing iron bar is at 800°C (1073 K). How much radiation does it emit per square meter?

Power/area = εσT⁴ = 0.7 × 5.67×10⁻⁸ × 1073⁴ = 0.7 × 5.67×10⁻⁸ × 1.327×10¹² = 52,600 W/m². For a bar with 0.01 m² surface area, that's 526 W — enough to noticeably heat a room.

Why does the T⁴ dependence matter so much?

Because small temperature increases cause dramatic radiation increases. A metal at 500 K radiates P ∝ 500⁴ = 6.25×10¹⁰. At 1000 K (doubled): P ∝ 10¹² — a 16× increase. This is why a dull red glow (800 K) is far less bright than white-hot (1500 K), which radiates ~25× more.

How do spacecraft manage their temperature in space?

With no air for convection, spacecraft rely entirely on radiation for cooling. Surfaces facing the Sun use reflective coatings (low absorptivity), while radiator panels have high emissivity coatings to dump waste heat into space. The James Webb Space Telescope's sunshield keeps instruments at −233°C this way.