surface brightness lecture

7/29/2019 Surface Brightness Lecture

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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


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.


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)!


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


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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• 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


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…


surface brightness lecture

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