Is redshift an accurate method for measuring distances of distant galaxies and quasars? I'm currently in 11th grade working on a science project about measuring distances using redshifts. I understand how the wavelength of a light wave increases when an object moves away from the point of observation. Now suppose a star, say 20 light years away, emits a light wave at this moment. Then that specific light wave must reach us after 20 years. If we use the redshift of this light wave to calculate the distance, then aren't we basically getting the position of the star as it was 20 years ago? Cause by this time the star has travelled much further away. Can someone please explain how the redshift-distance relation works and if I've gone wrong somewhere?
 A: I have found the perfect graph which I believe can explain the concepts.

Let’s suppose we have an object located at a comoving distance $r$. This objects emits light at $t=t_1$ (or $z$) and reaches us around $t=t_0$ (or $z=0$). By looking at this graph, we can define some distances.

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*Comoving distance, $r$
The comoving distance $r$, can be written as
$$r(z) = \frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$
As you can see from the graph, the comoving distance does not change with time.


*Proper distance, $d(z)$
The proper distance of an object can be written as,
$$d(z) = \frac{r}{1+z} \equiv \frac{1}{1+z}\frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$
In this case, from the graph, we can define 2 proper distances.
$$d_0(z) = r \equiv \frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$
$$d_1(z) = \frac{r}{1+z} \equiv \frac{1}{1+z}\frac{c}{H_0}\int_0^{z}\frac{dz'}{E(z')}$$
By setting $a(t)=1$, we see that the comoving distance at current instance is equal to the proper distance of the object.


*Conformal Time, $\eta(z)$
The conformal time can be defined as
$$\eta(z) = \frac{1}{H(z)}\int_0^z\frac{dz'}{E(z')}$$
which describes the distance traveled by the photon from $t_1$ to $t_0$.

If we use the redshift of this light wave to calculate the distance, then aren't we basically getting the position of the star as it was 20 years ago?

The redshift of the object and given cosmological model can help us to determine the distance of an object. In general there are 5-6 different distance definitions (such as angular diameter distance, luminosity distance and transverse comoving distance, lookback time). However, the basics are like this. So when you observe a galaxy and measure it's redshift you can obtain these distances.
