# AstroFormule

## Galaxies

### Instrumental resolution

Given spectra estimate instrumental resolution in AA and km/s:

$velocity\_scale[km/s] = \ln\left(\frac{\lambda_1}{\lambda_0}\right) \times speed\_of\_light$

$FWHM[AA] = 2.355 \times 5100 \times \frac{velocity\_scale}{speed\_of\_light}$

where λ01 are any two subsequent wavelengths in the equidistant spectrum. This is resolution at 5100AA. In the case of galaxies, FWHM should be divided with (1+redshift).

### 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

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

where DL is the luminosity distance.

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

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:

$log10(M_{HI}[M_{\odot}])=-0.4m_{21}^{0} + log10 (1+z) + 2 log10 (D[Mpc]) + 12.3364$,

where m210 is 21-cm line magnitude corrected for HI absorption and D is the luminosity distance. the absorption correction is: $m_{21}^{0} = m_{21} - A_{21}\ \textrm{and}\ A_{21} = 0.5 \times logR_{25}$, for Hubble type $T \ge 0$ and with $logR_{25}\equiv 1$ for $logR_{25}\ge 1$.

### 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 corrected to the centroid of the Local Group and for infall of the Local Group towards Virgo cluster, one may obtain luminosity distance using formula:

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

   $v_{LG}^{final}=v_{sys}+300*\sin(l)*\cos(b)$


where vsys is the radial velocity (cz in km/s) already corrected to the centroid of the Local Group.

IF DATA ARE TAKEN FROM HYPERLEDA DATABASE:

Having measured heliocentric radial velocity (v) as a weighted mean of optical and radio measurements vopt or vrad, respectively, one may calculate the velocity corrected to the centroid of the Local Group (denoted as vlg in the Hyperleda database):

  $v_{LG} = v\ +\ 295.4 * \sin(l) \cos(b)\ -\ 79.1* \cos(l)\cos(b)\ -\ 37.6* \sin(b)$


where (l,b) are Galactic coordinates, and could be obtained from equatorial using IDL routine glactc.

The velocity corrected for infall of the Local Group towards Virgo is noted vvir in the Hyperleda DB and here it is:

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


where 208km/s is the infall velocity of the Local Group (according to Theureau et al. 1998 and Terry et al. 2002) 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(sglo − sgl)


Equatorial coordinates could also be transformed to supergalactic using glactc.

So, to summarize, using Hyperleda database, one may calculate the luminosity distance simply taking: DL[Mpc] = vvir / H0, where H0 = 75km / sMpc − 1 is the Hubble constant.

### 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