Surface brightness

Surface brightness is a concept used in astronomy when describing extended astronomical objects such as galaxies and nebulae.

General description

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, while the star is so small it is effectively a point source in most observations, the galaxy may extend over several arcseconds or arcminutes. Therefore, the galaxy will be harder to see than the star against the airglow background light. Quoting an object's surface brightness gives an indication of how easily observable it is.

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 magnitude m extending over an area of A square arcseconds, the surface brightness S is given by:

$S=m+2.5\cdot \log A$

Surface brightness is constant with increasing object luminosity distance. For nearby objects, the luminosity distance is equal to the physical distance of the object. For a nearby object emitting a given amount of light, if it was twice as far away, a quarter of the intensity of light would reach us, but it would also have only a quarter the angular area, resulting in the same surface brightness.

Converting Surface Brightness in magnitude per square arcseconds to Solar Luminosity per square parsec

Formula

$S(mag/arcsec^{2})=M_{V,\odot }+21.57-2.5\log S(L_{\odot }/pc^{2})$

Where $M_{V,\odot }=$ Absolute Magnitude of the sun in V band (in our example)

NOTE: Other absolute magnitudes of the sun can be got from Galactic Astronomy or Absolute Magnitude of the Sun in Several Bands

Derivation

Assume that we are interested in finding the surface brightness of a galaxy at a distance d Mpc.

The luminosity enclosed within a box of 1 arcsec on each side is: $L_{V}\sim d^{2}\times a^{2}\times S(L_{\odot }/pc^{2})$

where $a={\frac {4.848\times d}{d}}{\frac {pc}{Mpc}}=4.848\times 10^{-6}$ for a galaxy (show it)

$S(mag/arcsec^{2})-M_{V,\odot }=-2.5\log L_{V}+5\log(d/10)$

substituting

$S(mag/arcsec^{2})-M_{V,\odot }=-2.5\log(d^{2}\times a^{2}\times S(L_{\odot }/pc^{2}))+5\log d+5$

expanding and simplifying

$S(mag/arcsec^{2})=M_{V,\odot }-2.5\log(S(L_{\odot }/pc^{2}))-5\log a-5$

Substituting for a, we get the required result.

Examples

(Problem 1.10 of Galaxies in the Universe: An Introduction): Show that the surface brightness $I_{B}=27mag/arcsec^{2}\sim 1L_{\odot }/pc^{2}$

Ans:

It is known that $M_{B,\odot }=5.48$. 

Substituting into the formula and re-arranging, we get $\log S(L_{\odot }/pc^{2})\sim 0$ which gives the required answer.

(Problem 5.2 of Galaxies in the Universe: An Introduction):

Show that the central surface brightness of 15 $mag/arcsec^{2}$ in the I band corresponds to $18000L_{\odot }/pc^{2}$

Ans:

$M_{I,\odot }=4.08$

Substituting in to the formula the answer is arrived at.

References

• Sparke, L.; Gallagher, J. (2000), Galaxies in the Universe: An Introduction (1st ed.), Cambridge University Press, ISBN 0-521-59241-0

See also

• Low surface brightness galaxy