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This question already has an answer here:

I have a catalog I am experimenting with that has some spectroscopic red shift data available (in the range from 0.5-1) as well as total Ks (K -short band) flux data available, how would I go about obtaining distance measures from Earth in parsecs or some other conventional units. Mainly I would need to know the method/formula for converting these values. Thanks!

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marked as duplicate by probably_someone, Sebastian Riese, Cosmas Zachos, glS, AccidentalFourierTransform Jun 12 '18 at 19:38

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  • $\begingroup$ I do not believe so, the answer provided uses transverse co-moving distance which is used mainly for 'two events at the same redshift, but separated on the sky by some angle' as said in David, H. (2000) Distance measures in cosmology. However correct me if there is a mistake in my understanding. $\endgroup$ – QuantumPanda May 25 '18 at 18:24
  • $\begingroup$ I would like to add that I am observing events in different red shifts. $\endgroup$ – QuantumPanda May 25 '18 at 18:33
  • $\begingroup$ There's a handy cosmological calculator that looks like it would do what you want at einsteins-theory-of-relativity-4engineers.com/cosmocalc.htm $\endgroup$ – D. Halsey May 25 '18 at 21:05
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For 0 < z < 20 we can neglect the radiation density in respect of the densities of matter and dark energy.

$$\Omega_{R_0}=0$$

We also assume flat universe.

$$\Omega_{K_0}=0$$

Then, the proper distance "d" can be calculated from de redshift "z" with:

$$\boxed{d=\frac{c}{H_0} \int_0^z \dfrac{dx}{\sqrt{\Omega_{M_0}(1+x)^3+\Omega_{\Lambda_0}}}}$$

The best values that we have for the parameters, come from Planck Collaboration 2015:

$$H_0=67.74 \ (km/s)/Mpc$$

$$\Omega_{M_0}=0.3089$$

$$\Omega_{\Lambda_0}=0.6911$$

$$c=299792.458 \ km/s$$

For each value of z the integral can be easily calculated using numerical methods

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  • $\begingroup$ How about for redshift z > 1? Would I need a different equation? $\endgroup$ – QuantumPanda May 25 '18 at 23:16
  • $\begingroup$ You can use this equation for z<20 with a relative error less than 0.1%, and for z<100 with a relative error less than 0.2% $\endgroup$ – Albert May 26 '18 at 8:57
  • $\begingroup$ Also, I noticed there is no constant, Omega subscript k , which is used on the specific model of the universe, ie curved, flat, etc. Does the above equation take the universe is flat (since Omega subscript k = 0) ? If so is this a valid assumption given my case ( 0<z<2) $\endgroup$ – QuantumPanda May 28 '18 at 14:46
  • $\begingroup$ Yes, the above equation assumes that the universe is flat, because that indicates the best measurements currently available. And if the universe is flat, it is flat for all redshifts, 0 < z < infinity. In other words, the above equation is the best that can be applied to your case. $\endgroup$ – Albert May 28 '18 at 20:18

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