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For an approximately blackbody star, the Stefan–Boltzmann relation connects luminosity, radius and effective temperature.
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Formula
R in m; L in W; σ=5.670374419·10−8 W/(m2·K4); Teff in K
| Symbol | Meaning | Unit |
|---|---|---|
| Radius of the approximately spherical stellar surface | m | |
| Bolometric total luminosity of the star | W | |
| Stefan–Boltzmann constant | W/(m2·K4) | |
| Teff | Effective temperature that gives the same total flux as a blackbody | K |
R=\sqrt{\frac{L}{4\pi\sigma T_{\mathrm{eff}}^4}}Solving L=4πR2σTeff4 for R gives this expression. Teff is an effective, not necessarily local, temperature; real stellar spectra depart from an ideal blackbody because of lines, atmosphere and extinction. Luminosity should be bolometric or corrected from band measurements with a model. Apparent brightness is not interchangeable with luminosity and also depends on distance.
Applies to an approximately spherical stellar surface, bolometric luminosity and effective temperature in a blackbody model; real atmospheres and extinction require corrections.
Multiply Stefan–Boltzmann flux per unit area by the stellar surface area, then solve for R.
Luminosity
Use the same blackbody and spherical model.
Temperature
The fourth root makes temperature relatively insensitive to small luminosity errors.
Compute the radius for L=3.828×1026 W and Teff=5772 K.
R=√(3.828×1026/(4π·5.6704×10−8·57724))≈6.96×108 m, about one solar radius.
At the same temperature, what happens if L rises by a factor of 9?
R is proportional to √L, so the radius triples.
Using apparent brightness as L.
L is total bolometric power; observed flux also depends on distance.
Substituting temperature in Celsius.
The radiation equation requires absolute temperature in kelvin.
Formula cluster
Effective temperature and bolometric luminosity jointly constrain the emitting stellar surface.
For L=3.828·1026 W and Teff=5772 K, R=√(L/(4πσTeff4))≈6.96·108 m, close to the solar radius.
Estimate a stellar radius from independently determined luminosity and effective temperature.
Here is how to work through a typical Stellar radius from luminosity and effective temperature (R = √(L/(4πσTeff⁴))) task step by step:
Task
Compute the radius for L=3.828×1026 W and Teff=5772 K.
Solution path
R=√(3.828×1026/(4π·5.6704×10−8·57724))≈6.96×108 m, about one solar radius.
Task
At the same temperature, what happens if L rises by a factor of 9?
Solution path
R is proportional to √L, so the radius triples.
The equation describes a specific relation among astronomical quantities; it supports a reliable conclusion only when measurements match the variables’ definitions. Start by identifying each quantity, unit and reference frame. The relation R=\sqrt{\frac{L}{4\pi\sigma T_{\mathrm{eff}}^4}} organizes those values mathematically, while the physical model determines whether it applies to the situation. A number alone does not establish that. Record assumptions, signs and any approximations. This keeps separate what follows directly from calculation and what requires interpretation. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
The relevant limits are stated in the subject background: Solving L=4πR2σTeff4 for R gives this expression. Teff is an effective, not necessarily local, temperature; real stellar spectra depart from an ideal blackbody because of lines, atmosphere and extinction. Luminosity should be bolometric or corrected from band measurements with a model. Apparent brightness is not interchangeable with luminosity and also depends on distance. Read them before substitution because ideal geometry, small angles, constant properties or a particular cause of motion may be part of the derivation. Measurement uncertainty is also distinct from model validity: a precise measurement does not make an unsuitable approximation correct. Check that quantities use consistent conventions. If the assumptions fail, use a more suitable model or explicitly report the result as an estimate rather than an exact measurement. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Use the documented worked example as a calculation check: For L=3.828·1026 W and Teff=5772 K, R=√(L/(4πσTeff4))≈6.96·108 m, close to the solar radius. Carry units through the arithmetic rather than handling numbers alone. Do they combine into the unit of the requested quantity? A dimensional mismatch often reveals a conversion error or swapped variables. If possible, substitute the result back into a rearranged original relation. Finally compare its order of magnitude with familiar astronomical scales; plausibility is not proof, but it is a useful error check. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Read the exponents and placement of quantities in R=\sqrt{\frac{L}{4\pi\sigma T_{\mathrm{eff}}^4}}: direct proportionality behaves differently from an inverse or power relation. Mentally double one input at a time and infer how the requested quantity changes. In observational data, a small relative uncertainty can matter greatly when a relation uses a small angle, a difference or an inverse quantity. Distinguish this sensitivity from measurement precision: the equation describes error propagation, not how well an instrument measures. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
State the numerical value with its unit, the input measurements and the main model assumption. If using an approximation, label the result as an estimate and say which effects were ignored. Avoid conclusions beyond the equation: a geometric or photometric quantity is not automatically a complete description of a star or galaxy. When measurement errors are known, do not report more precision than the data support. This concise record makes both calculation and interpretation traceable and reproducible. Compare the scale with the system being studied, and report uncertainty or systematic effects whenever they matter.
Quanta exam set
Curated exam set for "Stellar radius from luminosity and effective temperature":
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Which formula describes Stellar radius from luminosity and effective temperature?
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How do you rearrange Stellar radius from luminosity and effective temperature for Luminosity?
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Which common mistake happens with Stellar radius from luminosity and effective temperature?
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Create a curated FSRS exam set for R = √(L/(4πσTeff⁴)): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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