Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.astro.spbu.ru/staff/resh/Lectures/lec2.pdf Äàòà èçìåíåíèÿ: Mon Jan 26 21:19:23 2009 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:33:30 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: spiral galaxy

Photometric models of early-type galaxies de Vaucouleurs law

Photometric models of early-type galaxies de Vaucouleurs law

de Vaucouleurs law
General form of the de Vaucouleurs law: lg I () = - ( Ie
1/4

- 1),

where = r /re and ­ coefficient ( > 0). Let isophotes are homocentric ellipses with ellipticity = 1 - b/a. Then the total luminosity is
2 LT = 2 Ie re (1 - ) + 0

exp[- (

1/4

- 1)]d = 8!

e 2 (1 - )Ie re , 8

where = ln 10.

Photometric models of early-type galaxies de Vaucouleurs law

Photometric models of early-type galaxies de Vaucouleurs law

Growth curve of the galaxy is k () = L( ) = 1 - exp(- LT
1/4

Total (asymptotic) luminosity: )·
n =7 n =0

n n /4 . n!

2 2 LT = 7.21457 Ie re (1 - ) = 22.66523Ie re b/a.

Absolute magnitude: MVauc = µe - 5 lgre - 2.5 lg(1 - ) - 39.961, where the effective radius re is in kpc. Mean surface brightness within re is I e = 3.61Ie or µ
e

r = re ( = 1) k (1) = 1/2. Thus, = 7.66925 and = /ln 10 = 3.33071. Therefore, a final form of the de Vaucouleurs law is lg or, in units of m / I (r ) = -3.33071 Ie
1/4

r re

-1 ,

,

µ(r ) = µe + 8.32678[(r /re )1/4 - 1].

2 Total luminosity, expressed through I e , is LT = 2 I e re b/a. Central surface brightness of the de Vaucouleurs model is b I0 = 103.33 Ie = 2140Ie .

= µe - 1.39.


Photometric models of early-type galaxies de Vaucouleurs law

Photometric models of early-type galaxies de Vaucouleurs law

Major axis profile of NGC 3379 (solid line)

Dashed line ­ approximation with µe (B ) = 22.24 and re = 56. 8 (2.7 kpc). De Vaucouleurs law fits the s.b. profile within µ 10m with error ±0.m 08.

Growth curve for NGC 3379. Open circles ­ aper ture measurements, solid line ­ approximation by standard curve k () for the de Vaucouleurs law with BT = 10.20.

Photometric models of early-type galaxies de Vaucouleurs law

Photometric models of early-type galaxies de Vaucouleurs law

Deprojection of the de Vaucouleurs law
So far we have discussed observed surface brightness profile I (R), that is 3D distribution of light (stars) projected onto the plane of the sky. The question is whether we can, from this measured quantity, infer the real 3D distribution of light, j (r) in a galaxy. If I (R) is circularly symmetric, we can assume that j (r) will be spherically symmetric, and from the following figure it is apparent that:

This is an Abel integral equation for j as a function of I , and its solution is: j (r ) = - 1
r

dI dR . dR R 2 - r 2

3D density distribution:
(r ) = 0 r

assuming M/L = const (r ). exp(-r
1/4

Example of analytical approximation (Mellier & Mathez 1987):
-0.855

).

I (R ) = 2
R

-

dz j (r ) =
2

Therefore, where M0 = 16 0 (re / 4 )3 and Mtot = 1.65 · 104 M0 . M( r ) = M0 (8.58, r
1/4



j (r )rdr r 2 -R

.

),


Photometric models of early-type galaxies Sersic law

Photometric models of early-type galaxies Sersic law

Sersic law
Sersic profile is a generalization of the de Vaucouleurs profile: I (r ) = I0 e
-n
1/n

In units of m /



: 2.5n ln 10 r re
1/n

µ(r ) = µ0 + If n = 4

()
/4

,

where I0 ­ central surface brightness, = r /re , n > 0 and a constant n is chosen so that half the total luminosity predicted by the law comes from r re . Also, this profile can be written as I (r ) = exp -n Ie r re
1/ n

(*) µ(r ) = µe + 8.32678[(r /re )1

- 1].

Effective surface brightness for the Sersic law (µe = µ(re )) is µe = µ0 + 2.5n /ln 10. Luminosity within r : L( r ) = where ( , x ) =
x 0

-1

,

2 n (2n, n 2 n n

1/n

2 ) I 0 re ,

e -t t

-1

dt ­ incomplete gamma function.
2 n 2 (2n) I0 re , 2 n n

where Ie = I0 e-n . When n = 4 4 =7.66925 the Sersic law transforms to the de Vaucouleurs law.

Total (asymptotic) luminosity: LT =

where ( ) = ( , ) ­ gamma function.
Photometric models of early-type galaxies Sersic law

Photometric models of early-type galaxies Sersic law

Growth curve: k () =

L() LT

=

(2n,n 1/n ) (2n)

.

Sersic profiles for n = 1 - 10

Table: The values of n (Ciotti & Ber tin 1999) n 1 2 3 4 5 n 1.67834699 3.67206075 5.67016119 7.66924944 9.66871461 n 6 7 8 9 10 n 11.6683632 13.6681146 15.6679295 17.6677864 19.6676724

Analytical approximation (Ciotti & Ber tin 1999): 4 46 n = 2n - 1 + 405n + 25515n2 + O (n-3 ). 3

Luminous ellipticals, cD galaxies ­ n 4 or even 4, dwarf E ­ n 1.


Photometric models of early-type galaxies Other laws

Photometric models of early-type galaxies Other laws

Hubble-Reynolds formula
The first model used to describe the surface brightness profiles of elliptical galaxies (Reynolds 1913): 4I (r0 ) , I (r ) = (1 + r /r0 )2 (I (r ) r
-2

Modified Hubble law
(or modified Hubble-Reinolds law) I (r ) = at r >> r0 ) and
2 L( r ) = r0 ln[1 + (r /r0 )2 ].

I0 , 1 + (r /r0 )2

(I (r ) r

-2

at r >> r0 )

where r0 ­ characteristic radius of the distribution, I (r0 ) ­ surface brightness at r0 from the nucleus. Total luminosity of circular galaxy within r is L( r ) = 8 I (r0 )r
2 0 0

xdx 2 , = 8 I (r0 )r0 ln(1 + ) - 2 1+ (1 + x ) L( r ) .

Again, r L( r ) . Modified Hubble law corresponds to a simple analytical form for 3D distribution: j0 j (r ) = , [1 + (r /r0 )2 ]3/2 where j0 = I0 /2r0 .

where = r /r0 . As one can see, r

Photometric models of early-type galaxies Other laws

Photometric models of early-type galaxies Other laws

Hubble-Oemler law

King formula

I0 e I (r ) = (1 + r /r0 )2

-r /

2

rt2 -2

I (r ) = K [(1 + [r /rc ]2 )- where rc is the "core" radius ( and K ­ the scale factor.

1/2

- (1 + [rt /r ]2 )-

1/2 2

],

I (r =0) I (r =rc )

= 2), rt is the "tidal" radius

For r < rt the surface brightness changes as I (r ) r

.

For r > rt the surface brightness profile decays very quickly and predicts a finite total luminosity. In the limit rt this one reduces to the Hubble-Reynolds law.

This formula gives a very good representation of star counts in tidally-limited globular clusters and low-density spheroidal galaxies.

Jaffe law, Hernquist law

etc.


Photometric models of early-type galaxies Central regions of elliptical galaxies

Photometric models of early-type galaxies Central regions of elliptical galaxies

Centers of early-type galaxies
The HST observations of early-type galaxies reveal that the central par ts have surface brightness distibutions that are different from the extrapolation of traditional fitting formulae derived from ground-based observations.

The surface brightness profiles generally consist of two distinct regions: a steep power-law regime ­ I (r ) r - ­ at large radius, and a shallower power law ­ I (r ) r - ­ at small radius. Classification: < 0.3 ­ "core" galaxies (shallow inner slope), > 0.5 ­ power-law galaxies.

NGC 3115 (HST, F555W filter) µ = 0.m 44/

0 ­ core

Photometric models of early-type galaxies Central regions of elliptical galaxies

Photometric models of early-type galaxies Central regions of elliptical galaxies

Major-axis brightness profiles of Virgo ellipticals (V passband) ­ Kormendy (2009).

Surface brightness profiles for NGC 596 (open circles) ­ power-law nucleus, and NGC 1399 (solid circles) ­ galaxy with a core. Solid lines represent Nuker law fits (see fur ther).


Photometric models of early-type galaxies Central regions of elliptical galaxies

Standard models of disk galaxies

Nuker law
To parametrize the HST brightness profiles, Lauer et al. (1995) introduced general empirical double power law (the "Nuker" law): I (r ) = 2
-

Disk galaxies

I

b

rb r



1+

r rb



-

,

where , , , Ib , rb ­ parameters. The break radius, rb , is the radius at which the steep outer profile, I (r ) r - , "breaks" to become the inner shallow profile, I (r ) r - , and Ib = I (rb ). The Nuker law contains many simpler fitting formulae as special cases: The Hubble-Reynolds law corresponds to = 1, = 2, = 0; The modified Hubble law ­ = 2, = 2, = 0.

Standard models of disk galaxies Radial surface brightness distribution

Standard models of disk galaxies Radial surface brightness distribution

Radial distribution
Disks of spiral galaxies are known to show profiles described well by the "exponential law" (Patterson 1940, de Vaucouleurs 1959, Freeman 1970): I (r ) = I0 e or µ(r ) = µ0 + 1.0857 r /h, where h ­ exponential scale length, I0 or µ0 ­ central surface brightness of the disk. µ ­ r plane: exponential disk looks like straight line.
- r /h

Examples Shirley Patterson,
No. 914, pp.9-10, 1940

Harvard College Observatory Bulletin


Standard models of disk galaxies Radial surface brightness distribution

Standard models of disk galaxies Radial surface brightness distribution

Examples

Examples

NGC 300: exponential disk is traced up to 10h!
Kassin (2006)

Standard models of disk galaxies Radial surface brightness distribution

Standard models of disk galaxies Radial surface brightness distribution

Luminosity within r from the center L( r ) = 2 I0 h2 [1 - (1 + r /h)e total luminosity LT = 2 I0 h2 . Absolute luminosity of exponential disk Mexp = µ0 - 5 lgh - 38.57, where exponential scale length is in kpc. Growth curve k () = = r /h. L( ) = 1 - (1 + )e LT
- -r /h

],

, Aper ture photometry of M 33 (circles). Solid line is the growth curve for exponential disk with h = 9 and VT = 5.72.


Standard models of disk galaxies Radial surface brightness distribution

Standard models of disk galaxies Radial surface brightness distribution

Effective radius of exponential disk: re = 1.67835 h, effective surface brightness: Ie = I0 e-1.678 = 0.187 I0 or µe = µ0 + 1.822. In terms of effective parameters we can write total luminosity as 2 LT = 3.80332 Ie re . Mean surface brightness within effective radius is I e = 0.355 I0 or µ e = µ0 + 1.124.

Edge-on (i = 90o ) transparent disk: I (r ) = I
0

r r K1 , h h

where K1 is the modified Bessel function. r /h << 1: r /h >> 1: I (r ) I0 [1 + (r 2 /2h2 ) ln(r /2h)] I (r ) I0 r /2h e-r /h 1 + 8r3 /

h

Real stellar disks are not infinite. Exponential distribution typically extends out to about 5 radial scale lengths, beyond which disks are often truncated.

Standard models of disk galaxies Radial surface brightness distribution

Standard models of disk galaxies Ver tical structure of disks

Examples of truncated disks

Ver tical structure

UGC 11859 (B-band)

Standard model to describe ver tical surface brightness distribution in edge-on galaxies is isothermal self-gravitating sheet (e.g. van der Kruit & Searle 1981): I (z ) = I0 sech2 (z /z0 ),
de Grijs et al. (2001)

where z0 ­ ver tical scale (scale height).


Standard models of disk galaxies Ver tical structure of disks

Standard models of disk galaxies Ver tical structure of disks

Some galaxies demonstrate ver tical density profiles more sharply peaked near z = 0 than the sech2 (z /z0 ) model. Such data can be modelled better by exponential law: I (z ) = I0 e
van der Kruit & Searle (1981) -|z |/hz

,

where hz ­ exponential scale height. At z /z0 << 1 at z /z0 >> 1 and, therefore, approximately
2 sech2 (z /z0 ) = exp(-z 2 /z0 ), 2 sech (z /z0 ) = 4 exp(-2 z /z0 ) sech2 (z /z0 ) and exponential model give the same distribution with z0 = 2 hz .

In the framework of the model, ver tical scale z0 is connected with z (the dispersion in the velocities in the z -direction) and with (r , z ) (the space density of stars), (r ) (the projected density of stars):
2 2 z (r ) = 2 G(r , 0)z0 = G(r )z0 .

Ver tical velocity dispersion of an exponential disk is 2 z (r ) = 4 Ghz (r ) 1 - 1 e-|z |/hz . 2

Standard models of disk galaxies Ver tical structure of disks

Standard models of disk galaxies Ver tical structure of disks

van der Kruit (1988) proposed more general law (z ) = 2
-2/n

0 sech

2/n

3D disks
3D structure of disks: I (r , z ) = I (0, 0) e I (r , z ) = 0
- r /h

(nz /2z0 )

(n > 0).

The case n = 1 corresponds to the isothermal distribution (z ) = (0 /4) sech2 (z /z0 ), while the limiting case of n = is the exponential (z ) = 0 e-z /z0 .

sech2 (z /z0 )

(r rmax ) (r > rmax )

If i = 0o (face-on disk)
f I0ace -on +

= I (0, 0)
-

sech2 (z /z0 )dz = 2z0 I (0, 0).

For edge-on disk (i = 90o ) I Therefore, edge-on f I0 = I0ace
-on edge-on 0

= 2hI (0, 0). µ0
edge-on

h/z0

or

= µface 0

-on

h - 2.5 lg z0 .


Standard models of disk galaxies Ver tical structure of disks

Multi-component galaxies

For double exponential disk we have I (r , z ) = I (0, 0) e and µ0
edge-on -r /h-|z |/hz h - 2.5 lg hz .

= µface 0

-on

Therefore, for transparent disks the observed values of edge-on µ0 must be brighter than µface-on . 0 h/z0 5 for real galaxies µ 1.m 5 - 2m . Diameters of transparent edge-on disks must be larger than for face-on disks (measured within the same isophote). For instance, for typical disk with µface-on = 21.7 (B filter) and 0 h/z0 = 5 D26 (i = 90o ) 1.7. D26 (i = 0o )

NGC 5055 (g-band, mag.)

Real galaxies are multi-component systems: ­ bulge (de Vaucouleurs or Sersic law), ­ disk (exponential disk), ­ bar (e.g., Freeman's bar: Ibar (x , y ) = I0,bar 1 - (x /abar )2 - (y /bbar )2 ­ flattened elliptical disk)

Multi-component galaxies

Multi-component galaxies

Also: ­ lenses, ­ inner and outer rings, ­ spiral arms etc. Simplest case: two-component galaxy, consisting of de Vaucouleurs bulge and exponential disk. 2 Total luminosity: LT = Lbul + Ldisk = 2 (3.6073Ie re + I0 h2 ). Bulge-to-disk ratio: B /D = 3.6073 Growth curve: k (r ) = B /D k 1 + B /D
bul

Photometric decomposition of galaxies

Direct methods: analysis of 1D cuts or 2D images Software: GIM2D (Simard 1998), GALFIT (Peng et al. 2002), BUDDA (de Souza et al. 2004).

Ie I0

re h

2

.

(r ) +

1 k 1 + B /D

disk

(r ).


Multi-component galaxies

Multi-component galaxies

Other methods: ­ Iterative decomposition method proposed by Kormendy (1977), in which one solves for the disk parameters in a region where disk light dominates, and likewise for the bulge parameters. At each iteration, the light from the component being kept fixed is subtracted from the total surface brightness profile before the other component is solved for. The process is iterated until convergence is achived. ­ Kent (1986) presented a completely different approach: he made no assumption on the fitting laws for either component. He assumed that each one is characterized by elliptical isophotes of constant, and essentially different, flattenings. Then, an iterative process calculated the bulge and disk profiles. (Does not work for face-on galaxies ­ bulge and disk have roughly the same flattening.)

­ Colorimetric decomposition (statistical). Let the color index of the disk is KD , of the bulge ­ KB , and of the whole galaxy is KT . Then, 1 - 100.4(KD -KT ) B /D = - 1 - 100.4(KB -KT ) Example: normal Sa spiral galaxy with B - V = KT = +0.74, B - V = KD = +0.5 (disk), B - V = KB = +1.0 (bulge). Therefore, B /D = 0.73 (standard value for Sa galaxies is 0.68).