RED SHIT MAGNITUDE RELATION K correction
The magnitude of an oject is determined y m.5log const where is flux, which is related with asolute luminosity L of the oject as L 4l
So, we otain m m 5logl M ( z) m M M A is factor of extragalactic asorption and K is correction factor to a fixed andwidth K 5 A K.5log(1 z)
One can determine also the flux at given frequency, which is related with asolute luminosity at this frequency L n of the oject as L 4l where l n is monochromatic distance at this frequency
Suppose that the asolute luminosity is L The flux oserved at a frequency of n in the given frequency and, is emitted at a higher frequency and within a wider frequency and v o e /( 1 z) which together gives a correction in the form of a multiplier 1 z 1
In optical astronomy the introduction of K correction is much more difficult. That is ecause of presence the strong spectral lines, tilt of spectra, and wide optical ands. for the oserved flux and the square root of this factor for monochromatic distance l l (1 )/ 1 z That is K correction for extragalactic radiosources.
At high redshift, optical imaging samples the rest frame ultraviolet light in galaxies. Might this affect the oserved morphologies of high redshift galaxies? Could the unusual high redshift morphologies in the HD e due to the different rest frame wavelengths at high redshift? The answer is YES, and it is called sometimes morphological K correction.
The difference etween oserved and rest frame wavelengths changes the oserved flux of high redshift ojects, it is known as K correction. or instance, at summ and mm wavelength range galactic spectra are dominated y RJ slope of the thermal emission. The K correction is very strong and can make a distant galaxy righter then a closer identical galaxy.
SOURCE COUNTS The SOURCE COUNTS was one of the first test in cosmology. It also was related with so called Olers paradox. This paradox was formulated y the Dutch astronomer H.Olers. The first formulation was: if our Universe is infinite in space and time, then every line of sight should intersect with a star surface. So why the night sky is dark, not as right as our Sun?
Let us consider toy model of source distriution in cosmos. They are randomly and homogeneously distriuted over space and have equal luminosities. Apparent flux of the source which has distance r is L 4r Euclidean 3D space
So, the sources which have apparent flux greater than 0 are closer than r 0 : 1 0 0 4 L r The numer of sources which with > 0 is: 3 0 0 0 4 3 4 ) ( L n N where n 0 is average density of sources
One can differentiate N( 0 ) with respect to 0 and otain : d d lg N lg 0 3 One can introduce the density of sources n() per interval, + d: n( ) dn d
and : d lg n( ) d lg 5
Let us consider the mathematical properties of this distriution. irst of all it is very useful to calculate its mean, r.m.s. and other moments: 0 max min ) ( ) ( mean d n d n 1 max 1 min 3 0 5 3 0 1 1, ) ( n n n median median
when min 0 mean value It is Olers paradox, The mean value flux is diverges, the rightness of the sky ecomes infinite.
Let calculate r.m.s. 3 1 n( ) d n0median max 1 min
when max r.m.s. value rom it follows, that r.m.s. value is determined y the most powerful and closest source in the sky. The main conclusion of these results is: the sky should have the temperature of a source, T=6 000 0 K.
The resolution of this paradox is the evolution of the Universe The numer of sources is equal to product of source density n 0 y volume. The volume of a sphere is equal to V r( z) i.e. 4 3 π r ( z) where 3 is radius of comoving sphere, l cosmic distance r(z) 1 z
where l L 4 So, we can write N ( ) 4 n 3 3 L (1 z) 3 0 4
where L c H z 4 0 Sustituting this equation into N() we otain 1 0 3 0 4 3 1 4 3 4 ) ( L c H L n N
or d d lg N 3 3 H 0 lg c L 4
C.Pearson
aint lue galaxies were more numerous in the past, and may dominate the faint source counts