Comoving Distances and Their Meaning

Hubble's law:

\[v = H_0 \cdot d\]

Where:

- \(v\) is the velocity (in km/s)

- \(H_0\) is the Hubble constant (\(H_0 \approx 67.4\) km/s/Mpc)

- \(d\) is the distance (in Mpc)

For RS Puppis (\(d_1\)):

\[v_1 = H_0 \cdot d_1\]

\[d_1 = \frac{{v_1}}{{H_0}}\]

Using the redshift \(z_1\) for RS Puppis:

\[v_1 = H_0 \cdot z_1\]

\[v_1 \approx (67.4 \text{ km/s/Mpc}) \cdot (4.59 \times 10^{-41}) \approx 2.49 \times 10^{-39} \text{ km/s}\]

Now, find the distance in Mpc for RS Puppis:

\[d_1 \approx \frac{{v_1}}{{H_0}} \approx \frac{{2.49 \times 10^{-39} \text{ km/s}}}{{67.4 \text{ km/s/Mpc}}} \approx 3.70 \times 10^{-41} \text{ Mpc}\]

To convert this to light-years, using the fact that 1 Mpc is approximately equal to \(3.09 \times 10^{19}\) light-years:

\[d_1 \approx (3.70 \times 10^{-41} \text{ Mpc}) \cdot (3.09 \times 10^{19} \text{ light-years/Mpc}) \approx 1.14 \times 10^{-21} \text{ light-years}\]

So, the distance from Earth to RS Puppis (\(d_1\)) is approximately \(1.14 \times 10^{-21}\) light-years.

Now, let's calculate the distance from Earth to WHL0137-08 (\(d_2\)):

Using the redshift \(z_2\) for WHL0137-08:

\[v_2 = H_0 \cdot z_2\]

\[v_2 \approx (67.4 \text{ km/s/Mpc}) \cdot (9.11 \times 10^{-3}) \approx 0.61 \text{ km/s}\]

Now, find the distance in Mpc for WHL0137-08:

\[d_2 \approx \frac{{v_2}}{{H_0}} \approx \frac{{0.61 \text{ km/s}}}{{67.4 \text{ km/s/Mpc}}} \approx 0.009 \text{ Mpc}\]

To convert this to light-years:

\[d_2 \approx (0.009 \text{ Mpc}) \cdot (3.09 \times 10^{19} \text{ light-years/Mpc}) \approx 2.78 \times 10^{17} \text{ light-years}\]

So, the distance from Earth to WHL0137-08 (\(d_2\)) is approximately \(2.78 \times 10^{17}\) light-years.

Now that you have both distances in light-years:

- \(d_1\) (RS Puppis to Earth) \(\approx 1.14 \times 10^{-21}\) light-years

- \(d_2\) (Earth to WHL0137-08) \(\approx 2.78 \times 10^{17}\) light-years

You can find the distance from RS Puppis to WHL0137-08 by simply adding these two distances together:

Distance from RS Puppis to WHL0137-08 \(\approx d_1 + d_2\)

\(\approx (1.14 \times 10^{-21} \text{ light-years}) + (2.78 \times 10^{17} \text{ light-years}) \)

\(\approx 2.78 \times 10^{17} \text{ light-years}\)

So, the approximate distance from RS Puppis to WHL0137-08 is approximately \(2.78 \times 10^{17}\) light-years.