Useful Physical Constant in the c.g.s System
Nb: To write Greek symbols the Latex syntax is adopted. So, for instance:
\pi = Greek p = 3.14159
\sigma = the Greek sigma
Large or very small numbers are written in scientific notation. For istance, in this
notation 6.5e3 = 6500 and 3e-4 = 0.0003.
| Symbol | VALUE | UNITS | NAME |
|---|
| NA | 6.022e+23 | molecules/mole | Avogadro's number |
|---|
| k | 1.381e-16 | erg/oK | Boltzmann's constant |
|---|
| e | 4.803e-10 | e.s.u. | Electron charge |
|---|
| me | 9.109e-28 | g | Electron mass |
|---|
| G | 6.673e-8 | cm3/g/s2 | Gravitational constant |
|---|
| h | 6.625e-27 | erg/s | Planck constant |
|---|
| c | 2.998e+10 | cm/s | Speed of light |
|---|
| mp | 1.672e-24 | g | Proton mass |
|---|
| R=K NA | 8.314 | erg/mole/oK | Perfect Gas constant |
|---|
| \sigmaT | 6.652e-25 | (8\pi e4)/(3me2c4) | Thomson cross section |
|---|
| \sigma | 5.671e-5 | erg/s/cm2/oK4 | Stefan-Boltzmann constant |
|---|
Some Conversion Factors and Common Quantities
| mec2= | 8.186e-7 | erg | Electron energy at rest |
|---|
| 1 eV = | 1.602e-12 | erg | Electron Volt |
|---|
| 1 eV = | 11604 | oK | Electron Volt (Because E=KT) |
|---|
| 1 Joule= | 1e7 | erg | Energy units in MKS and cgs |
|---|
| 1 Tesla= | 1e4 | Gauss | Magnetic field unit in MKS and cgs |
|---|
| 1 Coulomb= | 3.0e6 | e.s.u. | Electric units conversion factor |
|---|
| 1 Jy = | 1e-23 | erg/s/cm2/Hz | Jansky |
|---|
| 1 pc = | 3.086e+18 | cm | Parsec |
|---|
| Msun= | 2e33 | g | Sun mass |
|---|
| Lsun= | 4e33 | erg/s | Sun luminosity |
|---|
| Rsun= | 6.96e10 | cm | Sun radius |
|---|
| AU = | 149.6e11 | cm | Astronomical Unit |
|---|
| LEdd= | (4\pi GMmpc)/\sigmaT | erg/s | Eddington luminosity |
|---|
| 1/H0= | 9.8e9 | years | Universe age for H0=100 km/s/Mpc |
|---|
| | 0.260 | eV/cm3 | Cosmic Microwave Background Energy density |
|---|
Effective Wavelength, Zeropoint and extinction
for Selected Photometric Systems
| Band | Eff. Wave. | mag zpt |
zero flux | Absorption |
|---|
| U | 3600 | -20.94 | 4.19e-9 | 1.57 |
|---|
| B | 4400 | -20.45 | 6.60e-9 | 1.34 |
|---|
| V | 5500 | -21.12 | 3.55e-9 | 1.00 |
|---|
| R | 6600 | -21.61 | 2.25e-9 | 0.75 |
|---|
| r | 6500 | -21.69 | 2.10e-9 | 0.75 |
|---|
| I | 8000 | -22.27 | 1.22e-9 | 0.48 |
|---|
| J | 12500 | -23.80 | 3.01e-10 | 0.28 |
|---|
| H | 16000 | -24.80 | 1.19e-10 | 0.19 |
|---|
| K | 21800 | -26.00 | 4.00e-11 | 0.11 |
|---|
| L | 34500 | -27.87 | 7.08e-12 | 0.06 |
|---|
Columns give:
1) Photometric band: Uppercase letters refer to the standard Johnson-Morgan/Cousins system based
on A0V stars. Lowercase letter to Schneider, Gunn & Hossel system.
2) Effective wavelength in Angstrom
3) Magnitude zeropoint. This is the magnitude of a source having a flux of 1 erg/s/cm^2/Angstrom.
The magnitude of a source with flux F is: m = -2.5Log(F) + zeropoint
4) Flux from a star having m = 0 ( erg/s/cm^2/Angstrom)
5) Interstellar extinction relative to the extinction in the V band.
Sources:
Magnitude zero point from Bessel 1990, PASP 102, 1181
Extinction from Cardelli Clayton & Mathis 1989, ApJ 345, 245
Physical Properties of Main-Sequence Stars
Stars spend most of their life "burning" hydrogen into helium, and
during this phase they have stable luminosity, temperature, and
size. The mass is the only parameter that determine all the
measurable properties of stars. For this reason, in the luminosity
versus temperature plane (the H-R diagram) the stars follow a well
defined narrow line called the main sequence. Stars in the main
sequence are called dwarf, even if they can be very large. The
following table gives the value of some useful parameters for main
sequence stars.
Columms give:
1) spectral class, temperature increases going from class M to class O.
The spectral type of the sun is G2.
2) Logarithm of the mass in solar mass units. The mass of the
sun is 2*1033 grams.
3) Logarithm of the luminosity in solar units. The luminosity of the
sun is 4*1033 erg/s.
4) Bolometric magnitude.
5) Visual magnitude.
6) Logarithm of the radius in solar units. The radius of
the sun is 6.96*108 m.
7) 8) 9) 10) Give the magnitude difference (the color) for varius bands
| Class | Lg(M/Msun) | Lg(L/Lsun) | Mbol |
MV | Lg(R/Rsun) | (U-B) | (B-V) | (V-R) | (V-I) |
|---|
| M6 | -1.0 | -2.9 | +12.1 | +15.5 | -0.9 | ... | ... | ... | ... |
|---|
| M5 | -0.8 | -2.5 | +10.9 | +13.9 | -0.56 | 1.22 | 1.61 | ... | 2.8 |
|---|
| M4 | -0.6 | -2.0 | +9.7 | +12.2 | -0.5 | ... | ... | ... | ... |
|---|
| M2 | -0.4 | -1.5 | +8.4 | +10.2 | -0.3 | ... | ... | ... | ... |
|---|
| M0 | -0.3 | -1.6 | +7.5 | +8.8 | -0.22 | 1.25 | 1.40 | ... | ... |
|---|
| K5 | -0.2 | -0.8 | +6.6 | +7.5 | -0.14 | 1.09 | 1.16 | 0.8 | 1.5 |
|---|
| K0 | -0.1 | -0.4 | +5.6 | +5.9 | -0.07 | 0.47 | 0.85 | 0.74 | 1.4 |
|---|
| G5 | -0.04 | -0.1 | +4.9 | +5.2 | -0.04 | 0.20 | 0.69 | ... | ... |
|---|
| G2 | +0.0 | +0.0 | +4.7 | +4.8 | +0.00 | 0.13 | 0.65 | 0.41 | 0.75 |
|---|
| G0 | +0.02 | +0.2 | +4.2 | +4.4 | +0.04 | 0.05 | 0.58 | 0.52 | 0.93 |
|---|
| F5 | +0.11 | +0.6 | +3.3 | +3.4 | +0.11 | 0.01 | 0.42 | ... | ... |
|---|
| F0 | +0.20 | +0.8 | +2.6 | +2.7 | +0.18 | 0.06 | 0.29 | 0.30 | 0.47 |
|---|
| A5 | +0.30 | +1.2 | +1.8 | +1.9 | +0.23 | 0.11 | 0.15 | ... | ... |
|---|
| A2 | +0.4 | +1.6 | +0.7 | +1.1 | +0.32 | ... | ... | ... | ... |
|---|
| A0 | +0.46 | +1.8 | +0.3 | +0.6 | +0.38 | -0.06 | 0.00 | 0.00 | 0.00 |
|---|
| B8 | +0.56 | +2.3 | -1.1 | -0.2 | +0.49 | -0.34 | -0.10 | ... | ... |
|---|
| B5 | +0.77 | +3.0 | -2.9 | -1.1 | +0.58 | -0.56 | -0.16 | ... | ... |
|---|
| B3 | +0.88 | +3.7 | -4.6 | -2.2 | +0.72 | -0.71 | -0.20 | ... | ... |
|---|
| B0 | +1.24 | +4.4 | -6.3 | 3.4 | +0.86 | -1.08 | -0.30 | ... | ... |
|---|
| O8 | +1.36 | +4.9 | -7.6 | -4.6 | +0.93 | -1.13 | -0.31 | ... | ... |
|---|
| O5 | +1.77 | +5.4 | -8.9 | -5.6 | +1.08 | -1.15 | -0.32 | ... | ... |
|---|
| O4 | +1.8 | +6.0 | -10.2 | -6.3 | +1.3 | ... | ... | ... | ... |
|---|
Source: ``Galactic Astronomy'' by Mihalas & Binney;
Ed. Freeman & Company
Galaxies Luminosity Function
The galaxy luminosity function F(L), gives the number of galaxies per Mpc3
( 1Mpc = 3.08*1024 cm) with luminosity
between L and L+dL. It is expressed combining
an exponential and a power law, obtaining the so called
Schechter function (after Schechter 1976).
F(L)dL = n* (L/L*)a exp(-L/L*) dL/L*
The value of the three parameters that enter into the luminosity function
depends on the class of galaxies considered and the wavelength (photometric
band) where the luminosity function in evaluated.
Some values valid for an Hubble constant of H0=100 km/s/Mpc are given in the
following table. For different values of H0, n* becames (H/100)3
times larger, and L* becames (H/100)2 smaller.
| Band | n* | a | L* | Reference |
|---|
| B | 0.016 | -1.25 | 1.2*1010 | Efstathious et al. 1988, MNRAS 232, 431 |
|---|
| V | 0.012 | -1.25 | 1.0*1010 | Kirshner et al. 1983, AJ 88, 1285 |
|---|
| R | 0.019 | -0.70 | 7.6*10 9 | Lin et al. 1996, ApJ 464, 60 |
|---|
NB. Luminosities are expressed in solar luminosities, which is 4.64*1025 W in the V band.
Abell classes for galaxy clusters
Cluster of galaxies have a wildly variable number of members, ranging
from just a few galaxies to several thousands. For each given cluster,
the total number of members is, however, not known because most
galaxies are too faint to be detectable at the telescope, and the
fraction of undetected galaxies becomes larger as more distant cluster
are considered. To cope with this observational limit, Abell has
introduced a measure of the cluster richness which is as independent
as possible on the cluster distance. The richness is defined as the
number of galaxies in the range m3 to m3+2,
where m3 is the magnitude of the third brightest member of
the cluster. Accordingly with this definition, cluster are divided in
6 classes of richness, given in the following table.
| Class | Population | | Class | Population |
|---|
| 0 | 30 - 49 | | 3 | 130 - 199 |
|---|
| 1 | 50 - 79 | | 4 | 200 - 299 |
|---|
| 2 | 80 - 129 | | 5 | >300 |
|---|