Planck Function Calculator

Calculator

Choose wavelength or frequency form. Compute a single point, or generate a table over a range for plotting and analysis.

Example Data Table

Sample values for a 3000 K blackbody using the wavelength form.

Your exact outputs depend on the chosen inputs and units.

Formula Used

Planck's law gives the spectral radiance of an ideal blackbody as a function of temperature and either wavelength or frequency.

Constants

• h = 6.62607015e-34 J*s
• c = 2.99792458e8 m/s
• k = 1.380649e-23 J/K

The calculator also reports Wien peak wavelength lambda_max = b/T and total exitance M = sigma*T^4, plus radiance I = sigma*T^4/pi.

How to Use This Calculator

1. Enter the blackbody temperature in kelvin.
2. Select wavelength form (Blambda) or frequency form (Bnu).
3. Choose “Single value” for one point, or “Range table”.
4. Provide the wavelength/frequency or range with units.
5. Press Submit to view results above the form.
6. Use CSV or PDF buttons to export your dataset.

Notes and Interpretation

• Blambda and Bnu describe the same spectrum but use different x-axes.
• Peak wavelength and peak frequency are not simple inverses.
• Use range tables for plotting smooth curves and comparing temperatures.
• Exitance sigma*T^4 is hemispherical power per unit area from a blackbody.
• Radiance sigma*T^4/pi is commonly used in thermal imaging models.

Professional Article

1) Why the Planck function matters

The Planck function describes how an ideal blackbody spreads radiant energy across wavelength or frequency at a given temperature. It supports detector background estimates, optical design tradeoffs, and thermal emission benchmarking for surfaces and scenes. In astronomy it helps compare stellar temperatures, while in imaging it links radiance models to temperature workflows.

2) Wavelength and frequency forms are not interchangeable

Bλ(T) is expressed per meter of wavelength, while Bν(T) is expressed per hertz of frequency. They describe the same spectrum but are different densities, so their values and peak locations differ. This is why the calculator evaluates either form directly instead of relying on peak inversion.

3) Temperature shifts the spectrum predictably

Wien’s displacement gives a quick peak estimate: λ_max ≈ 2.8978×10⁻³/T meters. At 300 K, the peak is near 9.66 µm (longwave infrared). At 1000 K, it moves to ~2.90 µm (shortwave infrared). At 5778 K, it sits near ~0.50 µm, within the visible range.

4) Radiometric totals provide useful checks

The Stefan–Boltzmann relation M = σT⁴ gives hemispherical exitance in W/m² and is a strong sanity check. At 300 K, M is about 459 W/m². At 1000 K, it rises to roughly 56.7 kW/m². The calculator also reports radiance I = σT⁴/π, widely used in imaging and calibration.

5) Numerical stability across extreme inputs

Planck’s law includes an exponential term that can overflow at short wavelengths or high frequencies. This tool uses guarded exponent evaluation to keep results stable. When x = hc/(λkT) or x = hν/(kT) becomes very large, radiance approaches zero in that band, consistent with physics.

6) Practical unit strategy for real projects

Use nm or µm for wavelength work and THz for frequency work so inputs stay readable. Internally, the calculator converts to SI (meters and hertz) before evaluation. Exporting a range table helps with plotting, temperature comparisons, and feeding radiative transfer or sensor models. It is also helpful for emissivity scaling and quick band-to-band comparisons in practice.

7) Interpreting the example table

For a 3000 K source, spectral radiance grows from the visible into the near infrared, then declines at longer wavelengths. That pattern explains why incandescent sources look warm: visible output exists, but substantial energy is emitted in infrared. Range mode shows where your bandpass lies relative to the peak.

8) Recommended workflow

Start with a single-point calculation at your band center to estimate magnitude. Next, generate a range table spanning your band, then export CSV for plotting and integration. Finally, compare σT⁴ totals across temperatures to keep your thermal model consistent.

FAQs

1) What does the calculator output represent?

It outputs spectral radiance for an ideal blackbody, either per wavelength (Blambda) or per frequency (Bnu). Values are computed in SI units after converting your chosen input units.

2) Why do the peak wavelength and peak frequency not match by inversion?

Because Blambda and Bnu are different spectral densities. Their peaks occur at different x values, so nu_peak is not simply c divided by lambda_peak.

3) Which mode should I use for thermal cameras?

Most thermal imaging bands are specified by wavelength, so Blambda is convenient. Use a wavelength range matching your bandpass and export the table for plotting or integration.

4) What is the difference between exitance and radiance?

Exitance σT^4 is total hemispherical power per area (W/m^2). Radiance σT^4/π is power per area per steradian (W/m^2·sr) for a Lambertian blackbody.

5) How many points should I choose for a range table?

For smooth plots, 31 to 101 points usually work well. Use more points for narrow spectral features or when you plan to numerically integrate over a sharp bandpass.

6) Can I use this for non-black surfaces?

Yes, as a baseline. Multiply the radiance by emissivity for a simple approximation, then account for reflections, transmission losses, and atmospheric absorption as needed for your application.

7) Why do I sometimes see extremely small values?

At short wavelengths or high frequencies, the exponential term becomes very large, driving radiance toward zero. That behavior is expected and indicates the band is far from the thermal peak.