Brightness temperature is the temperature that reproduces the observed brightness at a given frequency. For radio work, the Rayleigh–Jeans approximation is commonly used:
Tb,RJ = (c2 / (2 k ν2)) · Iν
If you start from flux density, the specific intensity is estimated from the solid angle Ω:
Iν = Sν / Ω
For a Gaussian source with FWHM axes θmaj, θmin:
Ω = (π / (4 ln 2)) · θmaj · θmin
The calculator can also invert the Planck function for higher-frequency cases:
Tb,Pl = (hν/k) / ln(1 + (2hν3) / (Iν c2))
Optional rest-frame reporting multiplies temperature by (1+z).
- Pick a computation method: flux and size or specific intensity.
- Enter either frequency or wavelength, then choose its unit.
- For flux-based inputs, provide flux density and angular axes (major and minor).
- Select an angular model to define the solid angle Ω.
- Optionally enter redshift and enable rest-frame scaling.
- Press Calculate. Use CSV/PDF buttons to export results.
| Case | Sν | ν | θmaj | θmin | Model | z | Expected outcome |
|---|---|---|---|---|---|---|---|
| Compact jet knot | 1 Jy | 5 GHz | 10 mas | 8 mas | Gaussian | 0.5 | High Tb, often 1010–1012 K |
| Resolved star-forming region | 50 mJy | 1.4 GHz | 2 arcsec | 2 arcsec | Disk | 0 | Lower Tb, commonly 102–105 K |
| Dust continuum map | 0.3 Jy | 350 GHz | 0.4 arcsec | 0.3 arcsec | Gaussian | 2 | Planck Tb may exceed RJ estimate |
- Beam vs source size: Use the deconvolved source size when available; using beam sizes can bias Tb.
- Gaussian factor: The π/(4 ln 2) term applies to FWHM axes for a 2D Gaussian.
- Planck vs RJ: At higher ν (shorter λ), Planck inversion can return higher Tb than RJ for the same brightness.
- Rest-frame scaling: The optional (1+z) multiplier is commonly used for extragalactic brightness temperatures.
1) What brightness temperature represents
Brightness temperature (Tb) converts measured sky brightness into an equivalent blackbody temperature at a chosen frequency. It does not require the source to be thermal; it is a convenient way to compare compactness and emission mechanisms across bands. Radio jets, star-forming regions, and masers often report Tb to assess beaming and optical depth.
2) Why frequency matters in practice
The Rayleigh–Jeans relation scales as ν−2 for fixed intensity, so the same map brightness implies a much lower Tb at higher frequency. For example, moving from 5 GHz to 50 GHz reduces the RJ temperature by a factor of 100 if Iν is unchanged. This is why multi-frequency imaging is essential for physical interpretation.
3) Typical values seen in observations
Thermal H II regions and free–free emission commonly sit around 102–104 K. Synchrotron sources in AGN cores often reach 109–1012 K in VLBI measurements. Very high Tb can indicate relativistic Doppler boosting, coherent processes, or unresolved structure.
4) Solid angle choices and their impact
The solid angle Ω is the dominant geometric factor when converting flux density to intensity. A Gaussian FWHM model uses Ω = (π/(4 ln 2)) θmaj θmin, which is larger than a simple disk using the same axes. If you underestimate θ by two, Tb increases by about four.
5) Planck inversion versus Rayleigh–Jeans
At millimeter and submillimeter bands (hundreds of GHz), hν can be comparable to kT. In this regime, Rayleigh–Jeans underestimates temperature for a given brightness, so Planck inversion is preferred. The calculator reports both, helping you quantify the divergence as frequency rises.
6) Rest-frame scaling with redshift
Many studies quote a rest-frame brightness temperature by multiplying the observed value by (1+z). This convention reflects the transformation of frequency and time dilation in cosmological sources. For a quasar at z = 2, the rest-frame reported Tb is three times the observed estimate.
7) Interferometric beam and deconvolution
If your source is not fully resolved, the fitted size may be close to the synthesized beam. Using beam dimensions instead of a deconvolved source size typically yields a lower limit on Tb. High signal-to-noise and proper model fitting are crucial when claiming extreme temperatures.
8) Quick sanity checks before publishing
Confirm units (Jy, mJy, MJy/sr), verify whether θ values are FWHM or diameters, and ensure the frequency matches your map. Compare the output against known benchmarks: the cosmic microwave background is about 2.725 K, while strong masers can exceed 1012 K. These checks prevent common reporting mistakes.
1) Is brightness temperature a real gas temperature?
No. It is an equivalent temperature that reproduces the observed brightness at a frequency. Non-thermal synchrotron or coherent emission can yield very high Tb without hot gas.
2) When should I prefer Planck inversion?
Use Planck inversion at high frequencies or short wavelengths where hν is not negligible compared to kT. This is common in mm/sub-mm continuum work and can significantly change inferred temperatures.
3) Should I use integrated flux or peak flux?
For resolved sources, integrated flux with the source solid angle is appropriate. For compact or beam-sized components, peak flux per beam may be used, but then Ω should match the component or beam model.
4) What angular size should I enter for interferometry?
Enter the deconvolved FWHM (or fitted component size) whenever possible. If you only have beam dimensions, the resulting Tb is typically a lower limit because the source may be smaller than the beam.
5) Why does Tb scale with (1+z) in some papers?
It is a reporting convention for rest-frame brightness temperature. Observed frequencies are redshifted, and multiplying by (1+z) produces a value comparable across sources at different redshifts.
6) Can Tb exceed 1012 K?
Yes, especially for masers, coherent bursts, or Doppler-boosted AGN cores. Extremely high values can also indicate that the source is unresolved or that the fitted size is underestimated.
7) Why are my results very sensitive to θ?
Because Ω is proportional to θmajθmin, and Tb is inversely proportional to Ω. Small changes in angular size can lead to large shifts in temperature, especially for compact sources.