# AstroFormule

### From MediaWiki

## Galaxies

### *Instrumental resolution*

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

where λ_{0},λ_{1} 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

where Mg_{sol} = 5.12, and m_{g} is the apparent visual (Petrosian) magnitude in g-band and R_{50} 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* *R*_{50}

where D_{L} is the luminosity distance.

### *Effective radius* *R*_{eff}

where *p*_{3} = 8 * 10^{ − 6} and *p*_{4} = 8.47

### *Inclination*

where 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 * *C*_{59}, where *C*_{59} is the concentration index.

### *Dynamical mass* from the width of HI line

where *W*_{20}[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^{ − 9}*M**p**c*(*k**m* / *s*)^{2} / *M*_{sol}.

### *Dynamical mass* in the old fashion way

-5<type<0 ::

type>0 ::

where *K*_{V} = 5 is the constant, *g* = 4.32 * 10^{ − 9}*M**p**c*(*k**m* / *s*)^{2} / *M*_{sol} the gravitational constant, *D*_{L} luminosity distance, and *V*_{s}, *V*_{r} stellar and rotational velocity given in km/s.

### *HI mass*

where *S*_{int} is the integrated HI flux and *D* is the luminosity distance.

Or using the prescription from the RC3 catalogue:

,

where *m*_{21}^{0} is 21-cm line magnitude corrected for HI absorption and *D* is the luminosity distance. the absorption correction is: , for Hubble type and with for .

### *True b/a ratio*

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

### *Luminosity* in g-band

where *M**g*_{sol} = 5.12, *m*_{g} the apparent visual magnitude and *D*_{L} the luminosity distance.

### *Gas fraction*

where M_{HI} 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:

- The classical IAU correction (Koribalski et al. 2004) for infall of the Local Group towards Virgo:

where *v*_{sys} 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):

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:

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 *s**g**l*,*s**g**b* in supergalactic coordinates and the direction of the center of the Virgo cluster (sglo = 104 deg,sgbo = −2 deg):

` cos(θ) = sin(`*s**g**b**o*)sin(*s**g**b*) + cos(*s**g**b**o*)cos(*s**g**b*)cos(*s**g**l**o* − *s**g**l*)

Equatorial coordinates could also be transformed to supergalactic using **glactc**.

**So, to summarize, using Hyperleda database, one may calculate the luminosity distance simply taking: D_{L}[Mpc] = vvir / H_{0}, where H_{0} = 75km / sMpc^{ − 1} is the Hubble constant.**

### *Maximum rotational velocity*

2*v*_{m}*s**i**n*(*i*) = (*W*_{20} + *W*_{50}) / 2
or if only one known, it is good as well.

IF TAKEN from HyperLeda, needs to be corrected for resolution (Paturel et al. 2003, A&A 412, 57):

*l**o**g*2*v*_{m}*s**i**n*(*i*) = *a* * *l**o**g**W*(*R*,*l*) + *b*; *l*=20,50, *R* = resolution

R = 8km/s: log 2 v_{m}sin(i) = (1.071 +/- 0.009) log W_{50}- (0.210 +/- 0.023) R = 8km/s: log 2 v_{m}sin(i) = (1.187 +/- 0.002) log W_{20}- (0.543 +/- 0.005) R =16km/s: log 2 v_{m}sin(i) = (1.048 +/- 0.003) log W_{50}- (0.156 +/- 0.006) R =16km/s: log 2 v_{m}sin(i) = (1.179 +/- 0.003) log W_{20}- (0.529 +/- 0.008) R =21km/s: log 2 v_{m}sin(i) = (1.049 +/- 0.002) log W_{50}- (0.158 +/- 0.005) R =21km/s: log 2 v_{m}sin(i) = (1.193 +/- 0.005) log W_{20}- (0.563 +/- 0.012) R =41km/s: log 2 v_{m}sin(i) = (1.052 +/- 0.009) log W_{50}- (0.176 +/- 0.021) R =41km/s: log 2 v_{m}sin(i) = (1.156 +/- 0.010) log W_{20}- (0.521 +/- 0.025)

If one has both W_{20} and W_{50}, 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*

where h=0.7 (Hubble constant), *g* and *r* apparent magnitudes and *D* distance in Mpc.

where *zp* is the zero point, dependent on the IMF. One may take *z**p* = − 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:

IMFOffsetReferenceKennicut 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,

If we use the restframe r-i color and L_{r} luminosity instead, then

These two estimates of M_{*} will differ because there is scatter in the (*g* − *r*) − (*r* − *i*) color plane.

### *Corrections*

- Velocity dispersion aperture correction

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

where *r*_{0} is the effective radius of the galaxy, and *r*_{ap} is the radius of the aperture. In the case of SDSS, the radius of the fiber is *r*_{ap} = 1.5".

- Magnitude (colors) corrections:

- Extinction correction

- K correction

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

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,K_{s},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:

with an initial value β=0.75

(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:

ev_{g}(z) = + 1.15zev_{r}(z) = + 0.85z