Astronomy:Mattig formula
Mattig's formula was an important formula in observational cosmology and extragalactic astronomy which gives relation between radial coordinate and redshift of a given source. It depends on the cosmological model being used and is used to calculate luminosity distance in terms of redshift.[1] It assumes zero dark energy, and is therefore no longer applicable in modern cosmological models such as the Lambda-CDM model, (which require a numerical integration to get the distance-redshift relation). However, Mattig's formula was of considerable historical importance as the first analytic formula for the distance-redshift relationship for arbitrary matter density, and this spurred significant research in the 1960s and 1970s attempting to measure this relation.
Without dark energy
Derived by W. Mattig in a 1958 paper,[2] the mathematical formulation of the relation is,[3]
[math]\displaystyle{ r_1 = \frac{c}{R_0 H_0} \frac{q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})}{q_0^2(1+z)} }[/math]
Where, [math]\displaystyle{ r_1=\frac{d_p}{R}=\frac{d_c}{R_0} }[/math] is the radial coordinate distance (proper distance at present) of the source from the observer while [math]\displaystyle{ d_p }[/math] is the proper distance and [math]\displaystyle{ d_c }[/math] is the comoving distance.
- [math]\displaystyle{ q_0=\Omega_0/2 }[/math] is the deceleration parameter while [math]\displaystyle{ \Omega_0 }[/math] is the density of matter in the universe at present.
- [math]\displaystyle{ R_0 }[/math] is scale factor at present time while [math]\displaystyle{ R }[/math] is scale factor at any other time.
- [math]\displaystyle{ H_0 }[/math] is Hubble's constant at present and
- [math]\displaystyle{ z }[/math] is as usual the redshift.
This equation is only valid if [math]\displaystyle{ q_0 \gt 0 }[/math]. When [math]\displaystyle{ q_0 \le 0 }[/math] the value of [math]\displaystyle{ r_1 }[/math] cannot be calculated. That follows from the fact that the derivation assumes no cosmological constant and, with no cosmological constant, [math]\displaystyle{ q_0 }[/math] is never negative.
From the radial coordinate we can calculate luminosity distance using the following formula,
- [math]\displaystyle{ D_L \ = \ R_0r_1(1+z) = \frac{c}{H_0q_0^2} \left[q_0z+(q_0-1)(-1+\sqrt{1+2q_0z})\right] }[/math]
When [math]\displaystyle{ q_0=0 }[/math] we get another expression for luminosity distance using Taylor expansion,
- [math]\displaystyle{ D_L = \frac{c}{H_0}\left(z+\frac{z^2}{2}\right) }[/math]
But in 1977 Terrell devised a formula which is valid for all [math]\displaystyle{ q_0 \ge 0 }[/math],[4]
- [math]\displaystyle{ D_L = \frac{c}{H_0}z\left[1+\frac{z(1-q_0)}{1+q_0z+\sqrt{1+2q_0z}}\right] }[/math]
References
- ↑ Observations in Cosmology, Cambridge University Press
- ↑ Mattig, W. (1958), "Über den Zusammenhang zwischen Rotverschiebung und scheinbarer Helligkeit", Astronomische Nachrichten 284 (3): 109, doi:10.1002/asna.19572840303, Bibcode: 1958AN....284..109M
- ↑ Bradley M. Peterson, "An Introduction to Active Galactic Nuclei", p. 149
- ↑ Terrell, James (1977), "The luminosity distance equation in Friedmann cosmology", Am. J. Phys. 45 (9): 869–870, doi:10.1119/1.11065, Bibcode: 1977AmJPh..45..869T
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