# Surface brightness

The overall brightness of an extended astronomical object such as a galaxy, star cluster, or nebula, can be measured by its total magnitude, integrated magnitude or integrated visual magnitude; a related concept is surface brightness, which specifies the brightness of a standard-sized piece of an extended object.

## General description

The total magnitude is a measure of the brightness of an extended object such as a nebula, cluster or galaxy. It can be obtained by summing up the luminosity over the area of the object. Alternatively, a photometer can be used by applying apertures or slits of different sizes diameter.[1] The background light is then subtracted from the measurement to obtain the total brightness.[2] The resulting magnitude value is the same as a point-like source that is emitting the same amount of energy.[3]

The apparent magnitude of an astronomical object is generally given as an integrated value—if a galaxy is quoted as having a magnitude of 12.5, it means we see the same total amount of light from the galaxy as we would from a star with magnitude 12.5. However, a star is so small it is effectively a point source in most observations (the largest angular diameter, that of R Doradus, is 0.057 ± 0.005 arcsec), whereas a galaxy may extend over several arcseconds or arcminutes. Therefore, the galaxy will be harder to see than the star against the airglow background light. Apparent magnitude is a good indication of visibility if the object is point-like or small, whereas surface brightness is a better indicator if the object is large. What counts as small or large depends on the specific viewing conditions and follows from Ricco's law.[4] In general, in order to adequately assess an object's visibility one needs to know both parameters.

## Calculating surface brightness

Surface brightnesses are usually quoted in magnitudes per square arcsecond. Because the magnitude is logarithmic, calculating surface brightness cannot be done by simple division of magnitude by area. Instead, for a source with a total or integrated magnitude m extending over a visual area of A square arcseconds, the surface brightness S is given by

$S = m + 2.5 \cdot \log_{10} A.$

For astronomical objects, surface brightness is analogous to photometric luminance and is therefore constant with distance: as an object becomes fainter with distance, it also becomes correspondingly smaller in visual area. In geometrical terms, for a nearby object emitting a given amount of light, radiative flux decreases with the square of the distance to the object, but the physical area corresponding to a given solid angle or visual area (e.g. 1 square arcsecond) increases in the same proportion, resulting in the same surface brightness.[5] For extended objects such as nebulae or galaxies, this allows the estimation of spatial distance from surface brightness by means of the distance modulus or luminosity distance.

## Relationship to physical units

The surface brightness in magnitude units is related to the surface brightness in physical units of solar luminosity per square parsec by

$S(mag/arcsec^2)=M_{\odot}+21.572-2.5\log_{10} S (L_{\odot}/pc^2),$

where $M_{\odot}$ and $L_{\odot}$ are the absolute magnitude and the luminosity of the Sun in chosen color-band[6] respectively.

## References

1. ^ Daintith, John; Gould, William (2006). The Facts on File dictionary of astronomy. Facts on File science library (5th ed.). Infobase Publishing. p. 489. ISBN 0-8160-5998-5.
2. ^ Palei, A. B. (August 1968). "Integrating Photometers". Soviet Astronomy 12: 164. Bibcode:1968SvA....12..164P.
3. ^ Sherrod, P. Clay; Koed, Thomas L. (2003). A Complete Manual of Amateur Astronomy: Tools and Techniques for Astronomical Observations. Astronomy Series. Courier Dover Publications. p. 266. ISBN 0-486-42820-6.
4. ^ Crumey, Andrew (2014). "Human contrast threshold and astronomical visibility". Monthly Notices of the Royal Astronomical Society 442: 2600. arXiv:1405.4209. Bibcode:2014MNRAS.442.2600C. doi:10.1093/mnras/stu992.
5. ^ Sparke & Gallagher (2000, § 5.1.2)
6. ^ Absolute magnitudes of the Sun in different color-bands can be obtained from Binney & Merrifield (1998) or Absolute Magnitude of the Sun in Several Bands

### General references

• Binney, James; Merrifield, Michael (1998). Galactic Astronomy. Princeton University Press. ISBN 978-0-691-02565-0
• Sparke, L.; Gallagher, J. (2000). Galaxies in the Universe: An Introduction (1st ed.). Cambridge University Press. ISBN 0-521-59241-0