Hubble's Law of Redshifts using observed distances

[Case_Files]

The problem statement, all variables, and given/known data:

Assume that the radial velocities v_r of galaxies, at the present time, are given by V_r=H₀*r, where H₀ = 65 km/(s*Mpc). However, we do not observe the present distances of galaxies, but the distances they had when light left them.

Plot the relation between radial velocity and distance that would be obtained directly from observations (i.e. the relation corresponding to measured distances, not present distances). Consider several values of distance, up to 2x10⁹ pc. Comment briefly on the shape of your curve.

relevant equations:

V_r=H₀*r

attempt at a solution:

I'm familiar with the shape of the graph showing Hubble's Constant, how radial velocity is proportional to the distance of the galaxy. I also understand, I think, that radial velocity is equal to the redshift z times c. What I'm really lost on is applying that to this question. We have a redshift -> recessional velocity because the galaxies are moving away, and that velocity/redshift is proportional to how far away that galaxy currently is. But I don't understand how this would change the relationship between the values/ the shape of the graph if we consider the measured/observed distances instead of their actual distances.

I'm not looking for someone to draw the graph for me, just to maybe help explain the question a little better and help me understand the information behind the answer.

 

[mark726]

Thank you for your post. It seems like you have a good understanding of the concept of radial velocity and how it is related to the redshift of galaxies. To answer your question about the shape of the graph, let's first consider the equation Vr=H0*r. This equation tells us that the radial velocity of a galaxy is directly proportional to its distance from us (r). In other words, the farther away a galaxy is, the faster it is moving away from us.

Now, when we observe the distances of galaxies, we are actually seeing the distances they had when the light left them, not their present distances. This means that the distances we observe are actually smaller than the present distances. So, if we plot the observed distances on the x-axis and the corresponding radial velocities on the y-axis, we would see a curve that is shifted downwards compared to the graph of present distances and radial velocities.

The shape of this curve would still be a straight line, but it would have a steeper slope compared to the graph of present distances and radial velocities. This is because the observed distances are smaller than the present distances, so the radial velocities would be higher for the same observed distance.

I hope this explanation helps you understand the concept better. Feel free to ask any further questions if you have them.