Steradian
SI derived unit · quantity: solid angle
01 · Definition
The steradian is the two-dimensional brother of the radian: it cuts out of a sphere not an arc but a patch. The unit is needed wherever radiation is distributed over directions — in photometry, in antenna theory, in detector physics.
One steradian is the solid angle that cuts out on a sphere of radius R an area equal to R squared. The full sphere contains 4π steradians, the hemisphere 2π, and a cone with an aperture of 65.54 degrees gives exactly one. The dimension, as with the radian, equals one: a square metre per square metre. It is through the steradian that the candela and the lumen are linked — luminous intensity multiplied by solid angle gives luminous flux, and without this unit photometry falls apart.
The definition speaks only of area, not of shape: a square patch with a side of one radius gives the same one steradian as a round cone. That is why in radio astronomy the solid angles of irregular regions of the sky are computed by plain integration.
02 · Conversion
Enter a solid angle — the sheet will convert it
Astronomers count areas on the sky in square degrees: the whole sky is 41 253, the constellation of Orion 594, the full Moon 0.2. A square degree is 3283 times smaller than a steradian, and it is the only practical alternative to it.
One steradian is about eight per cent of the full sphere. Hence a simple rule of thumb: a source of one candela shining in every direction gives 12.57 lumens, and within one steradian exactly one.
| Unit | Name | Value | Where it is met |
|---|---|---|---|
| sr | steradian, the SI unit | 1 | photometry, radiometry |
| msr | millisteradian | 1000 | LEDs, optics |
| µsr | microsteradian | 1 000 000 | lasers, antennas |
| nsr | nanosteradian | 1·10⁹ | astronomy |
| sphere | full sphere, 4π steradians | 0.07958 | an isotropic source |
| m²/m² | a square metre per square metre | 1 | the steradian is dimensionless |
| sq° | square degree | 3283 | sky surveys, catalogues |
| sq′ | square minute of arc | 1.182·10⁷ | telescope images |
| sq″ | square second of arc | 4.255·10¹⁰ | astrometry |
| hemisph. | hemisphere, 2π steradians | 0.15915 | a lamp above a table |
| sr | steradian | 1 | the present-day unit |
The square degree is the working unit of sky surveys: the whole sky is 41 253 square degrees. For narrow beams one takes square minutes and seconds of arc, and the hemisphere is named outright, because a lamp above a table shines into exactly that.
03 · Orders of magnitude
from a telescope pixel to the full sphereOne steradian — a cone of 65.5°
04 · Measuring instruments
Where steradians are counted
It measures luminous intensity by direction and integrates over steradians: that is how the full luminous flux in lumens is obtained from candelas. Without the steradian the integral would have nothing to be taken over.
The gain of an antenna is 4π divided by the solid angle of its beam. A narrow beam of a millisteradian gives a gain of about forty decibels — that is why a dish is aimed to within a fraction of a degree.
Every pixel collects light from its own solid angle. Sky brightness is measured in units per square arcsecond, and to convert into steradians one divides by 4.25·10¹⁰.
In particle physics the efficiency of a setup is expressed in fractions of 4π: a detector covering all directions catches every decay product.
05 · Writing rules
sr in lower case — Sr is strontium
The symbol is written with two lower-case letters; a capital S would give a chemical element. Like the radian, the steradian is dimensionless and may be omitted in formulas, but in writing the lumen it is kept: cd·sr reads unambiguously, whereas «candela» without sr gets confused with luminous intensity.
The square degree is not an SI unit but is permitted in astronomy. Prefixes are not applied to it: there is no such thing as a millisquare degree, and small areas are written in square minutes and seconds of arc.
06 · Neighbouring units
The steradian closes the pair of dimensionless SI units together with the radian and carries photometry on its back: a candela per steradian gives a lumen, and a lumen per square metre gives a lux. In radiometry the steradian defines radiance in watts per steradian square metre.
The Earth is seen from the Moon under a solid angle of about 0.0007 steradians — roughly thirteen times more than the Moon is seen from the Earth, because the Earth is that much larger across.
07 · Historical section
What solid angle was measured with
Area on the sky: the LSST survey covers 18 thousand, the whole sky 41 253. The standard of catalogues and surveys, kept in astronomy alongside the SI.
Before the steradian was introduced, lighting engineers described a beam by the diameter of the aperture and the distance. It had to be recalculated for every geometry, and results from different laboratories did not agree.
The efficiency of a detector is expressed not in steradians but as a percentage of 4π: that way it is clearer what share of the particles was caught and what share flew past unnoticed.
Instead of the solid angle of an antenna beam, the gain is given in decibels: 30 dBi means a beam of about 12 millisteradians. The engineer speaks in decibels, the formula in steradians.
The unit because of which the candela stayed a base one
Formally the steradian equals one, and in the 1970s a proposal was discussed to remove it from the system along with the radian, and to make the candela a derived unit of the watt. Lighting engineering spoke against it: without the steradian the distinction between luminous intensity and luminous flux disappears, and the whole body of industry documentation is built on that pair. The 1995 compromise gave both angular units the status of derived units with dimension one — formally they mean nothing, but in the notation cd·sr = lm they carry the sense.
Catalogue · the International System of Units
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