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Surface Brightness
Luminosity, L, is the radiative power (W ) emitted by a source. It is intrinsic to the source and thus
independent of distance. The quantity that is actually observed for galaxies, however, is the total intensity
I, which is the radiative power per unit area per solid angle (note that specific intensity, Iν , is the intensity
per unit frequency interval). The units of I are W m−2 ster−1 . When observing distant sources such as
galaxies, however, the solid angle subtended on the sky is usually small, so I is usually expressed in units
W m−2 arcsec−2 , where 1 arcsec = 1/60 arcmin = 1/3600 deg = π/(6.48 × 105 ) rad. For galaxies, I is
referred to as surface brightness – the total luminosity
emitted Rby a column of linear area dA.
R
Let us consider the solid angle dΩ = sin dφ sin θdθ ≈ 2π sin θdθ, which in the small-angle approximation (sin θ ∼ θ) valid for distant sources such as galaxies, reduces to dΩ ∼ πΘ2 . Same time θ ∝ 1/d
where d is the distance to galaxy. Then I ∝ d2LdΩ – approximately independent of d.
So, to a first approximation, I is independent of a galaxys distance d. This makes I an extremely useful
observational property of galaxies.
NOTE: this defines intensity only for a non-point type object.
The total luminosity L of a galaxy is difficult to define precisely for several reasons:
– galaxies do not have well-defined edges, and the extended regions are faint and have a low luminosity
gradient.
– light from a distant galaxy is redshifted by the expansion of the Universe (cosmological dimming of
1/(1 + z)4 ); this can be corrected for if d is known (which is usually not the case), but generally
causes higher z galaxies to have lower surface brightnesses. This is known as K-correction.
– the sky is very bright and unwanted light from other sources (listed in order of importance) must be
subtracted to deduce the surface brightness of a galaxy:
– zodiacal light : sunlight reflected off particulate matter in the Solar System
– air glow: photochemical processes in the upper atmosphere and city lights
– faint and unresolved stars in our galaxy
– diffuse extragalactic light from distant, unresolved galaxies
Much of the galaxy structure is fainter than the sky which must be accurately subtracted.
These problems are somewhat overcome by basing the magnitude M of a galaxy on the amount of light
emitted within a specified isophotal radius.
Isophotal radius at B = 25m defines the radius (if noted as R25) or diameter (if notes as D25) of the
galaxy. Historically connected to the limit on how much sky brightness can be subtracted (only if an object
has brightness of at least few per cent of the sky): sky has magnitude of 22m .6.
m1 − m2 = −2.5 log10
F1
2.3
F1
F1
⇒
= log10
⇒
=8
F2
2.5
F2
F2
That is best one can do when is not in space.
Surface brightness distributions
The absolute magnitude (or luminosity) of a galaxy tells us its total brightness. How the brightness is
distributed across the galaxy, i.e. the surface brightness distribution, gives us information about the internal
structure and dynamics of a galaxy. It is thus important to establish detailed models for how I varies with
radius R in a galaxy.
Ellipticals: are single component (spheroidal) systems. The best empirical fit for giant ellipticals and for
bright (MV < 17) ellipticals is the de Vaucouleurs R1/4 law:
1
"
#
1/4
h
i
R
I(R) = I(0) exp −kR1/4 = Ie exp −7.67
−1
Re
The effective radius, Re (measured along the semi-major axis a) is defined such that if the above law is
applied to all radii, then half the total light would be emitted interior to Re . So a plot of surface brightness
in magnitudes versus R1/4 gives a straight line for this law (since magnitude measures brightness on a
logarithmic scale).
Spirals and lenticulars: can be fitted by an exponential law:
R
I(R) = I(0) exp −
Rd
where Rd is a disk scale-length (typically ∼ 2 kpc). The central surface brightness of these galaxies is
approximately constant: I(0) ≈ 140L pc−2 . The surface brightness distribution of lenticulars follows a
spheroidal nucleus (like ellipticals) embedded in an exponential disk. In additional to the nuclear and disk
components, spirals also have a spiral arm component. S and SB types appear to have the same I(R) outside
the bar, but the bar itself appears to follow the R1/4 law typical of ellipticals.
Colour distribution
The colour of a galaxy is determined by the mix of stars in it. The colour distribution is obtained
by exposing specific colour-sensitive detectors to regions centred on a galactic nucleus and examining the
magnitude as a function of the aperture diameter, A. As A increases, light from regions further out from
the nucleus is included. The colours shift towards the blue (BV ,U B become more negative) as light from
more non-nuclear material is included. This is true of all galaxy types, except irregulars. For ellipticals, the
trend is only a few percent.
Ellipticals: The approximately constant colour distribution of ellipticals indicates that the same stellar
types dominate the luminosity in these systems. For example, giant ellipticals and dwarf ellipticals differ in
magnitude by many orders, but their colours are comparable: B − V ∼ 0.9 for giants and B − V ∼ 0.6 for
dwarfs.
Spirals and lenticulars: For lenticulars (S0) and spirals (S, SB), the colours of the spheroid and disk
components are different and differ with type. Nuclear regions are redder than disks. There is also a trend
towards bluer light in the outer disk most true in Sc galaxies. This must mean that the outer arms contain
younger stellar populations than the inner arms. Most of the light in spiral arms comes from early F to late
B giant stars. The average disk colour, B4 − V ∼ 0.15, is typical of late F/early G type stars. The average
colour of the spheroidal component, B − V <
∼ 0.8, is typical of early K types.
Luminosities The luminosity is related to M according to
L
= 10−(M −M )/2.5
L
where M = 4.74 is the absolute bolometric magnitude of the Sun and L ≈ 3.851026 W is the solar
luminosity.
Typical magnitudes and luminosities:
– giant ellipticals:M ≈ −22 =⇒ L ≈ 5 × 1010 L
– dwarf ellipticals: M ≈ −10 =⇒ L ≈ 8 × 105 L
– spirals and lenticulars: M ≈ −22to − 16 =⇒ L ≈ 5 × 1022 − 2 × 105 L
– (dwarf) irregulars M ≈ −18to − 12 =⇒ L ≈ 1 × 109 − 5 × 106 L
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