surface brightness lecture
TRANSCRIPT
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Galaxies – AS 3011 1
Lecture 5: Surface brightness and theSersic profile
• Surface brightness definition
• Surface brightness terminology
• The Sersic profile:
– Some useful equations
– Exponential profile
– De Vaucouleurs profile• Some examples:
– Profiles
– Equations
Surface brightness from AS1001
• Measure of flux concentration
• Units, mag/sq arcsec
• Range: 8-30 mag/sq/arsec
• High = compact = Low number!
•
Low = diffuse = High number!• Definition:
Distance independent in local Universe
Galaxies – AS 3011 2
Σ = I = f
4π R2
µ ∝−2.5log10( I ) [analag. to m∝−2.5log10( f )]
⇒ µ ∝m + 5log10( R)
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Measuring light profiles• NB, I (r) actually comes from an
azimuthal average, i.e of thelight between isophotes of thesame shape as the galaxy
• NB we do not always see thesteep rise in the core – if agalaxy has a small core, theatmospheric seeing may blur itout to an apparent size equal to
the seeing (e.g. 1 arcsec)• Often central profiles are
artificially flattened this way.
• Need to model PSF
azimuthal
angle φ
Example of a galaxy light profile
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The Sersic profile• Empirically devised by Sersic (1963) as a good fitting fn
I(r) = intensity at radius r
I0 = central intensity (intensity at centre)
α = scalelength (radius at which intensity drops by e-1)
n = Sersic index (shape parameter)
Can be used to describe most structures, e.g.,
Elliptical: 1.5 < n < 20 Bulge: 1.5 < n < 10
Pseudo-bulge: 1 < n < 2 Bar: n~0.5
Disc: n~1
Total light profile = sum of components.
I (r) = I 0 exp(−r
α
1n
)
Sersic shapes
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Connecting profile shape to total fluxThe best method for measuring the flux of a galaxy is to measure its
profile, fit a Sersic fn, and then derive the flux.
Galaxies – AS 3011 9
L =
0
2π
∫ I (r)rdr = 2π I 0
0
∞
∫ r exp(− rα
( )1
n
0
∞
∫ )dr
Substitute : x = ( rα
)1
n, r = xnα , dr = nx
n−1α
L = 2π I 0 xnα nx
n−1α
0
∞
∫ exp(− x)dx
L = 2π I 0α
2n x
2n−1
0
∞
∫ exp(− x)dx
Recognise this as the Gamma fn (Euler's integral or factorial fn):
Γ(z) = t z-1
0
∞
∫ e− t
dt = ( z −1)!
So : L = 2π I 0α 2
nΓ(2n)
Or for integer n : L = π I 0α
22n(2n −1)!= π I 0α
2(2n)!
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L = (2n)! π Σ0 α2
- Very important and useful formulae which connects the totalluminosity to directly measureable structural parameters.
- Most useful when expressed in magnitudes:
- For n=1:
- For n=4:
- So for fixed m as
- Note: SBs are like mags low value = high SB, high value=low SB !!
10−0.4m = (2n)!π 10−0.4µ
α 2
−0.4m = −0.4µ + log[(2n)!πα 2]
m = µ − 5log(α ) − 2.5log[(2n)!π ]
m = µ o− 5log10α − 2
m = µ o− 5log10α −12.7
α ↑µ o↑ I
o↓
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Q1) If a galaxy with a Sersic index of 1 has a measured SB
of µο=21.7 mag/sq arcsec, a scale-length of 2’’ and liesat a redshift of 0.1 what is its absolute magnitude ?[Assume Ho=100km/s/Mpc]
m = µ o − 5log10α − 2
m = 21.7 −1.5 − 2
m =18.2mags
z =v
c,v = H od => z =
H od
c
d =cz
H o= 300 Mpc
m −M = 5log10 d + 25
M = −19.2mags
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Q2) µο for an elliptical with n=4 is typically 15 mag/sq arcsec
at m=19.7 mags what α does this equate too ?
Too small to measure therefore one often redefines theprofile in terms of the half-light radius, Re, also known as
the effective radius.
m = µ o− 5log10α −12.7
α =100.2[µ
o−m−12.7]
α = 3.3×10−4 ' '
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Galaxies – AS 3011 13
• Scale-length to small to measure for high-n systems soadopt half-light radius (Re) instead.
• Can now sub for α and rewrite Sersic profile as:
• k can be derived numerically for n=4 as k=7.67
2π rI (r)dr =2π
20
Re
∫ rI (r)dr0
∞
∫
I (r) = I oexp(−[ r
( R
e
k n )]1n ) = I
oexp(−k [ r
Re
]
1
n )
I (r) = I oexp(−7.67[ R
Re
]14 )
2γ (2n,k ) = Γ(2n) = 2 t 2n−1
e−t dt
0
k
∫ = t 2n−1
0
∞
∫ e−t
, where : Re= k
nα , or, α =
Re
k n
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Therefore:
From previous example:
I.e., Re is measureable.
Central SB is also very difficult to get right for ellipticals asits so high, much easier to measure the surfacebrightness at the half-light radius. I.e.,
Re= 3459α
α = 3.3×10−4' '
Re=1.15''
I e= I
oexp(−7.67) = I
o10
−3.33
[Re :10− x = eln[10− x ]
= e− x ln10]
I (r) = I e103.3310
−3.33[ R Re
]
1
4
I (r) = I e10
−3.33[( R Re
)
1
4 −1]
I (r) = I eexp{−7.67[( R
Re
)1
4 −1]}
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This expression is known as the de Vaucouleur’s r 1/4 lawand is commonly used to profile ellipticals galaxies.
Note luminosity can be recast for n=4 as:
Recently the more generalised form with n free hasbecome popular with galaxies having a range of n from10-->0.5 (Note n=0.5 is a Guassian-like profile) as it fits
the variety of structures we see:
I (r) = I eexp{−k [( r
Re
)1n −1]}
L = π I oα
2(2n)!
For : n = 4, α =R
e
3459, I
0 =I e10
3.33
⇒ L = 7.2π I e R
e
2
".
Sab
c
In reality profiles are acombination of bulge plus
disc profiles with the
bulges exhibiting a Sersic
profile and the disc anexponential.
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Total light for 2 component system etc.• including the bulge, we can describe the profile as the
sum of a spiral and an elliptical power law:
Σ(R) = Σ0(d)
exp(-R/Rd) + Σe(b)
exp(-7.67 ([R/Re]1/4 -1))
• but if the bulge is rather flat (pseudo-bulge) we can justuse a second n=1 exponential with a smaller radius:
Σ(R) = Σ0(d)
exp(-R/Rd) + Σ0(b)
exp(-R/Rb)
• for the disk-plus bulge version, we get the totalluminosity
L = 2π Σ0(d)
Rd2 + 7.22π Re
2 Σe(b)
2 component system
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Bulge to disc ratio• a useful quantity to describe a spiral is the
bulge-to-total light ratio, which we can find
from the two luminosity terms
type B/T
S0 0.65
Sa 0.55
Sb 0.3
Sc 0.15
Size of a galaxy?
• Measuring the size of a galaxy is non-trivial
– Not clear where galaxy ends
– Some truncate and some don’t
– Need a standard reference
•
By convention galaxy sizes are specified by the half-lightradius (Re) or by scale-length (α)
• New quantitative classification scheme is based on thestellar mass versus half-light radius plane either for the
total galaxy or for components….work in progress…
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