AstroFormule

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Contents

Galaxies

Surface Brightness in g-band


\mu_g\left[\frac{L_{sol}}{pc^2}\right]=0.4 \left(Mg_{sol}+21.572-m_g-2.5*\log_{10}(2*\pi*R_{50}^2)\right),

where Mgsol = 5.12, and mg is the apparent visual (Petrosian) magnitude in g-band and R50 is Petrosian radius in arcseconds.

   M(g)= +5.12  (+/-0.02)
   u-g = +1.43  (+/-0.05)
   g-r = +0.44  (+/-0.02)
   r-i = +0.11  (+/-0.02)
   i-z = +0.03  (+/-0.02)

Calculating Surface Brightness in mag/arcmin squared
Calculate its area in square arc minute ( A=pi*a*b for elliptical object)
Example: Omega Centauri mag 3.7 dimension 36’ -> A=3.142*36*36 = 4069.44
SB= mag+2.5*log10(A) = 12.72 mag/sq arcmin

Petrosian radius R50


R_{50}[kpc] = R_{50}[''] \frac{1000 D_{L}[Mpc]}{206265},

where DL is the luminosity distance.


Effective radius Reff


R_{eff}[kpc] = R_{50}[''] \frac{ 1000 D_{L}[Mpc]}{206265} \left(1-p_3\frac{R_{90}['']}{R_{50}['']}\right)^{-p_4}= R_{50}[kpc]\left(1-p_3\frac{R_{90}['']}{R_{50}['']}\right)^{-p_4},

where p3 = 8 * 10 − 6 and p4 = 8.47


Inclination


\cos^2(i)=\frac{q^2-q_0^2}{1-q_0^2},\ q_0=0.37*10^{-0.053*t}\ (\ -5\le t\le 7)\ or\ q_0=0.42\ (\ t>7)

where q\equiv b/a is axes ratio (minor-to-major axis) and t is de Vaucouleurs morphological galaxy type.


  de Vaucouleurs(t)     -5   -3   -2   0   1   2   3   4   5   6   7   8   9   10     
        Type             E  E-S0  S0  S0a  Sa Sab  Sb Sbc  Sc Scd  Sd Sdm  Sm  Irr  

For a very imprecise determination of morphological type, one could use t = 10 − 2 * C59, where C59 is the concentration index.


Dynamical mass from the width of HI line


M_{dyn}=\frac{R_{50}['']D_L[Mpc]}{206265\ g} \left[\frac{W_{20}}{\sin(i)}\right]^2

where W20[km/s] is the velocity line width measured at 20% level of the peak flux, and i is the inclination. Gravitational constant g = 4.32 * 10 − 9Mpc(km / s)2 / Msol.


Dynamical mass in the old fashion way

-5<type<0 :: M_{dyn}=K_V \frac{(V_s^2+V_r^2)R_{50}['']D_L[Mpc]}{206265 g} \left(1-p_3\frac{R_{90}['']}{R_{50}['']}\right)^{-p_4}

type>0 :: M_{dyn}=\frac{V_r^2 R_{50}['']D_L[Mpc]}{206265 g} \left(1-p_3\frac{R_{90}['']}{R_{50}['']}\right)^{-p_4},

where KV = 5 is the constant, g = 4.32 * 10 − 9Mpc(km / s)2 / Msol the gravitational constant, DL luminosity distance, and Vs, Vr stellar and rotational velocity given in km/s.


HI mass

M_{HI}[M_{\odot}]=2.36*10^5\ D^2[Mpc]\ S_{int}[Jy\ km/s]

where Sint is the integrated HI flux and D is the luminosity distance.

Or using the prescription from the RC3 catalogue:

M_{HI}[M_{\odot}]=-0.4m_{21} + log10 (1+z) + 2 log10 (D[Mpc]) + 12.3364,

where m21 is 21-cm line magnitude and D is the luminosity distance.

M_{HI}/L_B = 1.5 \times 10^{-7} S_{int}[Jy\ km/s] 10^{0.4*(m_b-a_g)},

where mb is the apparent blue magnitude (= bt in LEDA) and ag estimated Galactic extinction.

  • Blue Luminosity in Lsun:

L_B = D^2 \times 10^{10 - 0.4 (mb - ag - MB,sun)} = 10^{0.4 (M_{B,sun} - M_{B,0})}

  • Absolute blue magnitude:

MB,0 = mbag − 5logD[Mpc] + 25

True b/a ratio


(b/a)_{true}=\frac{\sqrt{(b/a)^2-cos^2(i)}}{\sin(i)},

where b/a is the apparent minor-to-major axes ratio and i the inclination.

Luminosity in g-band


\log L_g = 0.4 \left(Mg_{sol}-m_g\right) - 2 + 2 \log_{10} \left(1000000 D_L[Mpc]\right)

where Mgsol = 5.12, mg the apparent visual magnitude and DL the luminosity distance.


Gas fraction


f_{gas} \equiv \frac{1.4 M_{HI}}{1.4 M_{HI} +M_*}

where MHI is the HI mass and M* is the stellar mass. The factor of 1.4 corrects for the mass in helium. The molecular gas component is neglected here (it is small in low-mass, LSB galaxies).


Luminosity distance

  • Knowing recession (systemic) velocity:


D[Mpc]=v_{LG}^{final}[km/s]/H_0

But first, we have to correct the systemic velocity to the centroid of the Local Group using

  • classical IAU correction
   vLG = vsys + 300 * sin(l) * cos(b)

IF DATA ARE TAKEN FROM HYPERLEDA DATABASE:

  • Paturel et al. 1998, A&A Suppl. Series, 124, 109
  v_{LG} = v_{sys}\ +\ 295.4* \sin(l) \cos(b)\ -\ 79.1* \cos(l)\cos(b)\ -\ 37.6* \sin(b)

where (l,b) are Galactic coordinates, and could be obtained from equatorial using IDL routine glactc; H0 = 70km / s / Mpc is the Hubble constant.

Second, we should correct vLG for the infall of the Local Group towards Virgo cluster:

  v_{LG}^{final} = v_{LG} + 208*cos(\theta)

where 208km/s is the infall velocity of the Local Group (Ho et al. 2007, ApJ 668, 94) and θ is the angular distance between the observed direction sgl,sgb in supergalactic coordinates and the direction of the center of the Virgo cluster (sglo = 104 deg,sgbo = −2 deg):

  cos(θ) = sin(sgbo)sin(sgb) + cos(sgbo)cos(sgb)cos(sglosgl)

Equatorial coordinates could also be transformed to supergalactic using glactc.

Maximum rotational velocity

2vmsin(i) = (W20 + W50) / 2 or if only one known, it is good as well.

IF TAKEN from HyperLeda, W_{20},\ W_{50} needs to be corrected for resolution (Paturel et al. 2003, A&A 412, 57):

log2vmsin(i) = a * logW(R,l) + b; l=20,50, R = resolution

  R = 8km/s: log 2 vm sin(i) = (1.071 +/- 0.009) log W50 - (0.210 +/- 0.023)
  R = 8km/s: log 2 vm sin(i) = (1.187 +/- 0.002) log W20 - (0.543 +/- 0.005)
  R =16km/s: log 2 vm sin(i) = (1.048 +/- 0.003) log W50 - (0.156 +/- 0.006)
  R =16km/s: log 2 vm sin(i) = (1.179 +/- 0.003) log W20 - (0.529 +/- 0.008)
  R =21km/s: log 2 vm sin(i) = (1.049 +/- 0.002) log W50 - (0.158 +/- 0.005)
  R =21km/s: log 2 vm sin(i) = (1.193 +/- 0.005) log W20 - (0.563 +/- 0.012)
  R =41km/s: log 2 vm sin(i) = (1.052 +/- 0.009) log W50 - (0.176 +/- 0.021)
  R =41km/s: log 2 vm sin(i) = (1.156 +/- 0.010) log W20 - (0.521 +/- 0.025)

If one has both W20 and W50, than they should both be corrected according to the table above, velocities extracted from the upper formula using inclination and then averaged.


Galaxy stellar mass


   logM_*[M_{\odot}]=1.097(g-r)-0.406-0.4\left[g+5-5log10(D[Mpc]*1000000)-5log10(h)-4.64\right]-2log10(h), 
where h=0.7 (Hubble constant), g and r apparent magnitudes and D distance in Mpc.


\log\left(\frac{M_{*,Bell}}{L_r}\right)_0 = 1.097 * (g - r)_0 - 0.406,\ {\rm since}\ \ \log \left(\frac{M_{*}}{L_{r}}\right)_0 = 1.097 (g-r)_0 + zp

where zp is the zero point, dependent on the IMF. One may take zp = − 0.306 + 0.15 − 0.25 = − 0.406. The standard diet-Salpeter IMF has zp = -0.306, which they state has 70% smaller M∗/Lr at a given color than a Salpeter IMF. In turn, a Salpeter IMF has 0.25 dex more M∗/Lr at a given color than the Chabrier (2003) IMF. Note that this expression requires luminosities and colors that have been k- and evolution-corrected to z=0.

Conversation table:

        IMF            Offset      Reference 
      Kennicut          0.30      Kennicut (1983) 
       Scalo            0.25      Scalo (1986)
    diet-Salpeter       0.15    Bell & de Jong (2001)
    pseudo-Kroupa       0.20      Kroupa (2001)
       Kroupa           0.30      Kroupa (2002)
      Chabrier          0.25     Chabrier (2003)
  Baldry & Glazbrook    0.305   Baldry & Glazbrook (2003)

Thus, in terms of restframe quantities,

 \log \left(\frac{M_{*,Bell}}{M_{\odot}}\right) = 1.097 * (g-r) -0.406 -0.4*(M_r-4.67)-0.19*z

If we use the restframe r-i color and Lr luminosity instead, then

 \log \left(\frac{M_{*,Bell}}{M_{\odot}}\right) = 1.097 * (r-i) -0.122 -0.4*(M_r-4.67)-0.23*z

These two estimates of M* will differ because there is scatter in the (gr) − (ri) color plane.

Corrections

  • Velocity dispersion aperture correction

Following Jørgensen, Franx, \& Kjaergaard (1995) and Wegner et al. (1999), we correct σest to a standard relative circular aperture defined to be one-eighth of the effective radius:

 
  \sigma_{corr}=\sigma_{est}\left(\frac{r_{ap}}{r_0/8}\right)^{0.04}
 

where r0 is the effective radius of the galaxy, and rap is the radius of the aperture. In the case of SDSS, the radius of the fiber is rap = 1.5".

  • Magnitude (colors) corrections:
Extinction correction


K correction

Using values from Bernardi et al. (2003a) for SDSS galaxies:

 K_g(z) = - 5.261\ z^{1.197}   
K_r(z) = - 1.271\ z^{1.023}

in g- and r-band respectively. For many different surveys (SDSS, GALEX, 2MASS, DEEP2, GOODS) and many bands (U,B,V,R,I,J,H,Ks,u,g,r,i,z), it can be obtained using Blanton's kcorrect code.

Cosmological dimming correction 

Jørgensen et al. (1995a):

C(z) =  − (1 + z)4 
Evolution correction
  • Driver et al. 2005MNRAS, 360, 81

Global form:  L(z) = L(z=0) \ (1+z)^{\beta}

with an initial value β=0.75

  E(z) = -0.75 \times 2.5 \log(1+z)  

(See Phillipps and Driver 1995MNRAS, 274, 832)

Bernardi et al. (2003b) reported that the more distant galaxies in their sample are brighter than those nearby:

evg(z) =  + 1.15z
evr(z) =  + 0.85z
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