Star Luminosity Calculator: L = 4 pi R^2 sigma T^4

The total power a star radiates, from its size and surface temperature.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose which quantity you want to find.

Enter the other values in the units shown.

The calculator gives the answer with the formula, the substituted numbers, and the answer in other units.

Formulas used

  • L = 4\pi R^2\sigma T^4
  • T = \left(\frac{L}{4\pi R^2\sigma}\right)^{1/4}

Worked example

With the default numbers in the calculator above:

  1. The formula, arranged to find the luminosity: luminosity = 4 x pi x sigma x radius of the star^2 x surface temperature^4
  2. Put the values in (every value in its SI unit): luminosity = 4 x pi x sigma x 6.957e+8^2 x 5772^4
  3. The answer: luminosity: 3.82799e+26 W
  4. The same answer in other units: 0.999998 solar luminosities

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want to find? Surface temperature; Luminosity (leave it as it is if this is what you are finding) 3.8268E+26 W; Radius of the star (leave it as it is if this is what you are finding) 695700000 m.

  1. The formula, arranged to find the surface temperature: surface temperature = ((luminosity) / (4 x pi x sigma x radius of the star^2))^0.25
  2. Put the values in (every value in its SI unit): surface temperature = ((3.8268e+26) / (4 x pi x sigma x 6.957e+8^2))^0.25
  3. The answer: surface temperature: 5771.55 K

Common questions

How do I find the luminosity of a star?

Multiply the surface area (4 pi R squared) by sigma times the fourth power of the surface temperature.

What is the luminosity of the Sun?

About 3.83e26 watts.

Why does a small hot star outshine a large cool one?

Because the power rises with the fourth power of temperature but only with the square of radius.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any astronomy or physics text (blackbody radiation, stellar luminosity).

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